Apparatus and method for transmitting and/or receiving data over a fiber-optical channel employing perturbation-based fiber nonlinearity compensation in a periodic frequency domain

ABSTRACT

An apparatus for determining an interference in a transmission medium during a transmission of a data input signal according to an embodiment has a transform module configured to transform the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to obtain a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient is assigned to one of the frequency channels, an analysis module configured to determine the interference by determining one or more spectral interference coefficients, wherein each spectral interference coefficient is assigned to one frequency channel. The analysis module is configured to determine each spectral interference coefficient depending on the spectral coefficients, and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the argument values indicates one frequency channel, and wherein the transfer function is configured to return a return value depending on the argument values.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of copending InternationalApplication No. PCT/EP2019/067484, filed Jun. 28, 2019, which isincorporated herein by reference in its entirety, and additionallyclaims priority from European Application No. 19181865.7, filed Jun. 21,2019, which is also incorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

The present invention relates to an apparatus and method fortransmitting and/or receiving data over a channel, in particular, to anapparatus and method for transmitting and/or receiving data over afiber-optical channel employing perturbation-based fiber nonlinearitycompensation in a periodic frequency domain.

In communication theory, discrete-time end-to-end channel models play afundamental role in developing advanced transmission and equalizationschemes. Most notable the discrete-time linear, dispersive channel withadditive white Gaussian noise (AWGN) is often used to modelpoint-to-point transmission scenarios. In the last decades, a largenumber of transmission methods matched to such linear channels haveemerged and are now applied in many standards in the field of digitaltransmission systems. With the advent of high-speed CMOS technology,those schemes have also been adopted in applications for fiber-opticaltransmission with digital-coherent reception [1]. However, many of theapplied techniques (e.g., coded modulation, signal shaping andequalization) are designed for linear channels whereas the fiber-opticalchannel is inherently nonlinear. An exact model to obtain the outputsequence from a given input sequence by an explicit input/outputrelation is highly desirable to make further advances in developingstrategies optimized for fiber-optical transmission.

Indeed, many works in the past two decades were devoted to developchannel models for fiber-optic transmission with good trade-offs betweencomputational complexity and numerical accuracy. Starting from thenonlinear Schrodinger equation (NLSE), approximate solutions can beobtained following either a perturbative approach (cf. [2, P. 610]) orthe equivalent method of Volterra series transfer function (VSTF) (cf.[3], [4]). These channel models can approximate the nonlineardistortion—there commonly termed nonlinear interference (NLI)—up to theorder of the series expansion of the NLSE. A comprehensive summary ofrecent developments on channel models can be found in [5, Sec. I].

One particular class of channel models—based on a first-ordertime-domain perturbative approach—has been published in the early 2000sin a series of contributions by Antonio Mecozzi in collaboration with agroup from AT&T Labs [6]-[8]. The results, however, were limited totransmission schemes that were practical at that time (e.g.,dispersion-managed transmission, intensity-modulation anddirect-detection) and the details of the theory and its derivation werepublished only recently in [9]. A follow-up seminal paper with Rene-JeanEssiambre [10] extents the former work by including the matched filterand T-spaced sampling after ideal coherent detection. One central resultof this work is the integral formulation of the Volterra kernelcoefficients providing a first-order approximation of theper-modulation-interval T equivalent end-to-end input/output relation.Based on this work the joint contributions with Ronen Dar and colleagues[11]-[1 3] resulted in the so-called pulse-collision picture of thenonlinear fiber-optical channel. Here, the properties of cross-channelNLI were properly associated with certain types of pulse collisions intime -domain.

SUMMARY

According to an embodiment, an apparatus for determining an interferencein a transmission medium during a transmission of a data input signalmay have: a transform module configured to transform the data inputsignal from a time domain to a frequency domain comprising a pluralityof frequency channels to obtain a frequency-domain data signalcomprising a plurality of spectral coefficients, wherein each spectralcoefficient of the plurality of spectral coefficients is assigned to oneof the plurality of frequency channels, and an analysis moduleconfigured to determine the interference by determining one or morespectral interference coefficients, wherein each of the one or morespectral interference coefficients is assigned to one of the pluralityof frequency channels, wherein the analysis module configured todetermine each of the one or more spectral interference coefficientsdepending on the plurality of spectral coefficients and depending on atransfer function, wherein the transfer function is configured toreceive two or more argument values, wherein each of the two or moreargument values indicates one of the plurality of frequency channels,and wherein the transfer function is configured to return a return valuedepending on the two or more argument values.

According to another embodiment, a method for determining aninterference in a transmission medium during a transmission of a datainput signal may have the steps of: transforming the data input signalfrom a time domain to a frequency domain comprising a plurality offrequency channels to obtain a frequency-domain data signal comprising aplurality of spectral coefficients, wherein each spectral coefficient ofthe plurality of spectral coefficients is assigned to one of theplurality of frequency channels, and determining the interference bydetermining one or more spectral interference coefficients, wherein eachof the one or more spectral interference coefficients is assigned to oneof the plurality of frequency channels, wherein determining each of theone or more spectral interference coefficients is conducted depending onthe plurality of spectral coefficients and depending on a transferfunction, wherein the transfer function is configured to receive two ormore argument values, wherein each of the two or more argument valuesindicates one of the plurality of frequency channels, and wherein thetransfer function is configured to return a return value depending onthe two or more argument values.

Another embodiment may have a non-transitory digital storage mediumhaving stored thereon a computer program for performing the aboveinventive method for determining an interference in a transmissionmedium during a transmission of a data input signal, when said computerprogram is run by a computer.

An apparatus for determining an interference in a transmission mediumduring a transmission of a data input signal according to an embodimentis provided. The apparatus comprises a transform module configured totransform the data input signal from a time domain to a frequency domaincomprising a plurality of frequency channels to obtain afrequency-domain data signal comprising a plurality of spectralcoefficients, wherein each spectral coefficient of the plurality ofspectral coefficients is assigned to one of the plurality of frequencychannels. Moreover, the apparatus comprises an analysis moduleconfigured to determine the interference by determining one or morespectral interference coefficients, wherein each of the one or morespectral interference coefficients is assigned to one of the pluralityof frequency channels. The analysis module configured to determine eachof the one or more spectral interference coefficients depending on theplurality of spectral coefficients and depending on a transfer functionwherein the transfer function is configured to receive two or moreargument values, wherein each of the two or more argument valuesindicates one of the plurality of frequency channels, and wherein thetransfer function is configured to return a return value depending onthe two or more argument values.

A method for determining an interference in a transmission medium duringa transmission of a data input signal according to an embodiment isprovided. The method comprises:

-   -   Transforming the data input signal from a time domain to a        frequency domain comprising a plurality of frequency channels to        obtain a frequency-domain data signal comprising a plurality of        spectral coefficients, wherein each spectral coefficient of the        plurality of spectral coefficients is assigned to one of the        plurality of frequency channels. And:    -   Determining the interference by determining one or more spectral        interference coefficients, wherein each of the one or more        spectral interference coefficients is assigned to one of the        plurality of frequency channels.

Determining each of the one or more spectral interference coefficientsis conducted depending on the plurality of spectral coefficients anddepending on a transfer function, wherein the transfer function isconfigured to receive two or more argument values, wherein each of thetwo or more argument values indicates one of the plurality of frequencychannels, and wherein the transfer function is configured to return areturn value depending on the two or more argument values.

Moreover, computer programs are provided, wherein each of the computerprograms is configured to implement one of the above-described methodswhen being executed on a computer or signal processor.

In embodiments, it is aimed to complement the view on T-spacedend-to-end channel models for optical transmission systems by anequivalent frequency-domain description. The time discretizationtranslates to a 1/T-periodic representation in frequency.

Remarkably, the frequency-matching which is imposed along with thegeneral four wave mixing (FWM) process is still maintained in theperiodic frequency domain. The structure of this paper is organized asfollows. The notation is briefly introduced and the system model ofcoherent fiber-optical transmission is presented. Starting from thecontinuous-time end-to-end relation of the optical channelanintermediate result following the perturbation approachthe discrete-timeend-to-end relation is derived. We particularly highlight the relationbetween the time and frequency representation and point out theconnection to other well-known channel models. The relevant systemparameters, i.e., memory and strength, of the nonlinear response areidentified which lead to design rules of practical schemes for fibernonlinearity mitigation. For such schemes, a novel algorithm in1/T-periodic frequency-domain is introduced well-suited also for systemsoperating at very high symbol rates. Similar to the pulse-collisionpicture, certain degenerate mixing products in frequency domain can beattributed to a pure phase and polarization rotation. This in turnmotivates the extension of the original regular perturbation model to acombined regular-logarithmic model taking the multiplicative nature ofcertain distortions properly into account. The theoreticalconsiderations are complemented by numerical simulations which are inaccordance with results obtained by the split-step Fourier method(SSFM). Here, the relevant metric to assess the match between bothmodels is the mean-squared error (MSE) between the two T-spaced outputsequences for a given input sequence.

A discrete-time end-to-end fiber-optical channel model based on afirst-order perturbation approach is provided. The model relates thediscrete-time input symbol sequences of co-propagating wavelengthchannels to the received symbol sequence after matched filtering andT-spaced sampling. To that end, the interference from both self- andcross-channel nonlinear interactions of the continuous-time opticalsignal is represented by a single discrete-time perturbative term. Twoequivalent models can be formulatedone in the discrete-time domain, theother in the 1/T-periodic continuous-frequency domain. The time-domainformulation coincides with the pulse-collision picture and itscorrespondence to the frequency-domain description is derived. Thelatter gives rise to a novel perspective on the end-to-end input/outputrelation of optical transmission systems. Both views can be extendedfrom a regular, i.e., solely additive model to a combinedregular-logarithmic model to take the multiplicative nature of certaindegenerate distortions into consideration. An alternative formulation ofthe GN-model and a novel algorithm for application in low-complexityfiber nonlinearity compensation are provided. The derived end-to-endmodel entails only a single computational step and shows good agreementin a mean-squared error sense compared to the incremental split-stepFourier method.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be described below in moredetail with reference to the appended drawings, in which:

FIG. 1a illustrates an apparatus for determining an interference in atransmission medium during a transmission of a data input signalaccording to an embodiment;

FIG. 1b illustrates another embodiment, wherein the apparatus furthercomprises a signal modification module and an inverse transform module;

FIG. 1c illustrates a further embodiment, wherein the apparatus furthercomprises a signal modification module and an inverse transform module;

FIG. 1d illustrates another embodiment, wherein the apparatus furthercomprises a transmitter module configured to transmit the correctedtime-domain data signal over the transmission medium;

FIG. 1e illustrates a further embodiment, wherein the apparatus furthercomprises a receiver module configured to receive the data input signalbeing transmitted over the transmission medium;

FIG. 2 illustrates a generic fiber-optical transmission system model;

FIG. 3a illustrates a transmitter frontend of a generic fiber-opticaltransmission system model;

FIG. 3b illustrates an optical channel of the generic fiber-opticaltransmission system model;

FIG. 3c illustrates an receiver frontend of the generic fiber-opticaltransmission system model and variables associated with the regularperturbation model;

FIG. 4a illustrates definitions of variables in the time-domain;

FIG. 4b illustrates definitions of variables in the frequency-domain;

FIG. 5 illustrates a magnitude in logarithmic scale of a single-spannonlinear transfer function according to an embodiment;

FIG. 6 illustrates a magnitude in logarithmic scale of a single-spannonlinear transfer function according to another embodiment;

FIG. 7a illustrates a contour plot for a single-channel, single-span,lossless fiber scenario in the regular time-domain model according to anembodiment;

FIG. 7b illustrates a contour plot for a single-channel, single-span,lossless fiber scenario in the regular-logarithmic model according toanother embodiment;

FIG. 8a illustrates an energy of the kernel coefficients in time-domainover the symbol rate according to an embodiment;

FIG. 8b illustrates an energy of the kernel coefficients in afrequency-domain over the symbol rate according to a further embodiment;

FIG. 9a illustrates a contour plot in the regular model in the frequencydomain of the normalized mean-square error in dB for a single-channel,single-span, lossless fiber according to an embodiment;

FIG. 9b illustrates a contour plot in the regular-logarithmic model inthe frequency domain of the normalized mean-square error in dB for asingle-channel, single-span, lossless fiber according to an embodiment;

FIG. 10a illustrates contour plots of the normalized mean-square errorσ_(e) ² in dB according to an embodiment, wherein the results areobtained from the regular-logarithmic time-domain model over a standardsingle-mode fiber with end-of-span lumped amplification, and wherein thesymbol rate and the optical launch power are varied for single-spantransmission having a fixed roll-off factor;

FIG. 10b illustrates contour plots of the normalized mean-square errorσ_(e) ² in dB according to an embodiment, wherein the results areobtained from the regular-logarithmic time-domain model over a standardsingle-mode fiber with end-of-span lumped amplification, and wherein theroll-off factor and number of spans N_(sp) are varied with fixed symbolrate having fixed launch power;

FIG. 11a illustrates an energy of the kernel coefficients in thetime-domain;

FIG. 11b illustrates kernel energies Eh for cross-channel interference(XCI) imposed by a single wavelength channel;

FIG. 12a illustrates a contour plot of the normalized mean-square errorin dB in a time domain, for dual-channel, single-span, lossless fiberaccording to an embodiment; and

FIG. 12b illustrates a contour plot of the normalized mean-square errorin dB in a frequency domain, for dual-channel, single-span, losslessfiber according to another embodiment.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 a illustrates an apparatus for determining an interference in atransmission medium during a transmission of a data input signalaccording to an embodiment.

The apparatus comprises a transform module 110 configured to transformthe data input signal from a time domain to a frequency domaincomprising a plurality of frequency channels to obtain afrequency-domain data signal comprising a plurality of spectralcoefficients (A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃], . .. ), wherein each spectral coefficient of the plurality of spectralcoefficients (A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃], . . . ), isassigned to one of the plurality of frequency channels.

Moreover, the apparatus comprises an analysis module 120 configured todetermine the interference by determining one or more spectralinterference coefficients (e.g., ΔA_(λ) ^(SCI)[μ]), wherein each of theone or more spectral interference coefficients (e. g., ΔA₂ ^(SCI)[μ]) isassigned to one of the plurality of frequency channels.

The analysis module 120 configured to determine each of the one or morespectral interference coefficients (e. g., ΔA_(λ) ^(SCI)[μ]) dependingon the plurality of spectral coefficients (A_(λ)[μ], A_(λ)[μ₁],A_(λ)[μ₂], A_(λ)[μ₃], . . . ), and depending on a transfer function(H_(ρ)[μ₁, μ₂, μ₃]; H_(v)(ω₁, ω₂, ω₃)) wherein the transfer function(H_(ρ)[μ₁, μ₂, μ₃]: (ω₁, ω₂, ω₃)) is configured to receive two or moreargument values (μ₁, μ₂, μ₃; ω₁, ω₂, ω₃), wherein each of the two ormore argument values (μ₁, μ₂, μ₃; ω₁, ω₂, ω₃) indicates one of theplurality of frequency channels, and wherein the transfer function isconfigured to return a return value depending on the two or moreargument values (μ₁, μ₂, μ₃; ω₁, ω₂, ω₃).

In an embodiment, the transmission medium may, e.g., be a fiber-opticalchannel.

FIG. 1b illustrates another embodiment, wherein the apparatus furthercomprises a signal modification module 130 being configured to modifythe frequency-domain data signal using the one or more spectralinterference coefficients to obtain a modified data signal. Theapparatus of FIG. 1 b further comprises an inverse transform module 135configured to transform the modified data signal from the frequencydomain to the time domain to obtain a corrected time-domain data signal.

According to an embodiment, the signal modification module 130 of FIG. 1b may, e.g., be configured to combine each one of the one or morespectral interference coefficients (e.g., ΔA_(λ) ^(SCI)[μ]), or a valuederived from said one of the one or more spectral interferencecoefficients (e.g., ΔA_(λ) ^(SCI)[μ]), and one of the plurality ofspectral coefficients (A_(λ)[μ], A_(λ)[₁], A_(λ)[μ₂], A_(λ)[μ₃], . . . )to obtain the modified data signal. In a particular embodiment, thesignal modification module 130 of FIG. 1b may, e.g., be configured tocombine each one of the one or more spectral interference coefficients(e.g., ΔA_(λ) ^(SCI)[μ]), or a value derived from said one of the one ormore spectral interference coefficients (e.g., ΔA_(λ) ^(SCI)[μ]), andone of the plurality of spectral coefficients (A_(λ)[μ], A_(λ)[μ₁],A_(λ)[μ₂], A_(λ)[μ₃], . . . ) to obtain the modified data signal bysubtracting

-   -   each one of the one or more spectral interference coefficients        (e.g., ΔA_(λ) ^(SCI) [μ]), or a value derived from said one of        the one or more spectral interference coefficients (e. g.,        ΔA_(λ) ^(SCI)[μ]),    -   from one of the plurality of spectral coefficients (A_(λ)[μ],        A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃]), . . . );

or, in another embodiment, to obtain the modified receivesignal/sequence by subtracting

-   -   each one of the one or more spectral interference coefficients        (e. g. , ΔA_(λ) ^(SCI)[μ]), or a value derived from said one of        the one or more spectral interference coefficients (e. g.,        ΔA_(λ) ^(SCI)[μ]),    -   from one of the plurality of the spectral coefficients of the        distorted receive sequence Y_(λ)[μ];

or, in a futher embodiment, to inverse Discrete Fourier Transform thespectral interference coefficients ΔA_(λ)[k] to obtain time domaininterference coefficients Δa_(λ)[k], and to subtract the time domaininterference coefficients Δa_(λ)[k] from the (time-domain) receivesequence Y_(λ)[k];

or, in a yet further embodiment, to obtain the modified data or receivesignal by subtracting

-   -   each one of the one or more spectral interference coefficients        (e. g. , ΔA_(λ) ^(SCI)[μ]), or a value derived from said one of        the one or more spectral interference coefficients    -   (e. g., ΔA_(λ) ^(SCI)[μ]), and by multiplying each one of the        one or more spectral phase and polarization coefficients        (exp(−jϕ_(λ)I−j{right arrow over (S)}_(λ)))    -   from one of the plurality of spectral coefficients (A_(λ)[μ],        A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃], . . . ) or from one of the        plurality of the spectral coefficients of the distorted receive        sequence Y_(λ)[μ].

In an embodiment, the transform module 110 of FIG. 1 b may, e.g., beconfigured to transform the data input signal from the time domain tothe frequency domain by transforming a plurality of overlapping blocksof the data input signal from the time domain to the frequency domain toobtain a plurality of blocks of the frequency-domain data signal. Theinverse transform module 135 may, e.g., be configured to transform themodified data signal from the frequency domain to the time domain bytransforming a plurality of blocks from the frequency domain to the timedomain and by overlapping said plurality of blocks being represented inthe time domain to obtain the corrected time-domain data signal.

FIG. 1c illustrates a further embodiment, wherein the apparatus furthercomprises an inverse transform module 135 configured to transform theone or more spectral interference coefficients (e. g., ΔA_(λ) ^(SCI)[μ])from the frequency domain to the time domain. The apparatus of FIG. 1cfurther comprises a signal modification module 130 being configured tomodify the data input signal being represented in the time domain usingthe one or more spectral interference coefficients being represented inthe time domain to obtain a corrected time-domain data signal.

According to an embodiment, the signal modification module 130 of FIG.1c may, e.g., be configured to combine each one of the one or morespectral interference coefficients being represented in the time domain,or a value derived from said one of the one or more spectralinterference coefficients, and a time domain sample of a plurality oftime domain samples of the data input signal being represented in thetime domain to obtain the corrected time-domain data signal.

In a particular embodiment, the signal modification module 130 of FIG.1c may, e.g., be configured to combine each one of the one or morespectral interference coefficients being represented in the time domain,or a value derived from said one of the one or more spectralinterference coefficients, and a time domain sample of a plurality oftime domain samples of the data input signal being represented in thetime domain to obtain the corrected time-domain data signal bysubtracting

-   -   each one of the one or more spectral interference coefficients        being represented in the time domain, or a value derived from        said one of the one or more spectral interference coefficients,    -   from a time domain sample of a plurality of time domain samples        of the data input signal being represented in the time domain.

In an embodiment, the transform module 110 of FIG. 1c may, e.g., beconfigured to transform the data input signal from the time domain tothe frequency domain by transforming a plurality of overlapping blocksof the data input signal from the time domain to the frequency domain toobtain a plurality of blocks of the frequency-domain data signal. Theinverse transform module 135 may, e.g., be configured to transform aplurality of interference coefficients blocks from the frequency domainto the time domain, said plurality of blocks comprising the one or morespectral interference coefficients (e.g., ΔA_(λ) ^(SCI)[μ]). The signalmodification module 130 may, e.g., be configured to modify theoverlapping blocks of the data input signal, being represented in thetime domain, using the plurality of interference coefficients blocks toobtain a plurality of corrected blocks, wherein the signal modificationmodule 130 is configured to overlap the plurality of corrected blocks toobtain the corrected time-domain data signal.

FIG. 1d illustrates another embodiment, wherein the apparatus furthercomprises a transmitter module 140 configured to transmit the correctedtime-domain data signal over the transmission medium.

FIG. 1e illustrates a further embodiment, wherein the apparatus furthercomprises a receiver module 105 configured to receive the data inputsignal being transmitted over the transmission medium.

In an embodiment, the analysis module 120 may, e.g., be configured todetermine an estimation of a perturbated signal depending on the datainput signal using the one or more spectral interference coefficients(e.g., ΔA_(λ) ^(SCI)[μ]).

According to an embodiment, the analysis module 120 may, e.g., beconfigured to determine the estimation of the perturbated signal byadding each one of the one or more spectral interference coefficients(e.g., ΔA_(λ) ^(SCI)[μ]) with one of the plurality of spectralcoefficients (A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[₃], . . .).

In an embodiment each of the two or more argument values may, e.g., be achannel index (μ₁, μ₂, μ₃) being an index which indicates one of theplurality of frequency channels.

Or, in another embodiment, each of the two or more argument values is afrequency (ω₁, ω₂, ω₃) which indicates one of the plurality of frequencychannels, wherein said one of the plurality of frequency channelscomprises said frequency.

In an embodiment, the analysis module 120 may, e.g., be configured todetermine each spectral interference coefficient (e.g. ΔA_(λ) ^(SCI)[μ])of the one or more spectral interference coefficients (e. g., ΔA_(λ)^(SCI)[μ]) by determining a plurality of addends. The analysis module120 may, e.g., be configured to determine each of the plurality ofaddends as a product of three or more of the spectral coefficients(A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃], . . . ) and of the returnvalue of the transfer function, the transfer function having three ormore channel indices or three or more frequencies as the two or moreargument values of the transfer function, which indicate three or moreof the plurality of frequency channels to which said three or more ofthe spectral coefficients (A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃], .. . ) are assigned.

In an embodiment, the analysis module 120 may, e.g., be configured todetermine each spectral interference coefficient (e. g., ΔA_(λ)^(SCI)[μ]) of the one or more spectral interference coefficients (e.g.,ΔA_(λ) ^(SCI)[μ]) by determining a plurality of addends, wherein theanalysis module 120 may, e.g., be configured to determine each of theplurality of addends as a product of three or more of the spectralcoefficients (A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₂], A_(λ)[μ₃], . . . ) and ofthe return value of the transfer function, the transfer function havingthree or more channel indices or three or more frequencies as the two ormore argument values of the transfer function, which indicate three ormore of the plurality of frequency channels to which said three or moreof the spectral coefficients (A_(λ)[μ], A_(λ)[μ₁], A_(λ)[μ₁], A_(λ)[μ₂],A_(λ)[μ₃], . . . ) are assigned.

According to an embodiment, the analysis module 120 may, e.g., beconfigured to determine each spectral interference coefficient (e. g.,ΔA_(λ) ^(SCI)[μ]) according to:

${\Delta\;{A_{\lambda}^{SCI}\lbrack\mu\rbrack}} = {{- j}\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}} \times {\sum_{\mu_{1},\mu_{2}}{{A_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{A_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$

wherein ΔA_(λ) ^(SCI)[μ] is said spectral interference coefficient,wherein A_(λ)[μ₁] is a first one of the three or more spectralcoefficients, wherein A_(λ)[μ₂] is a second one of the three or morespectral coefficients, wherein , A_(λ)[μ₃] is a third one of the threeor more spectral coefficients, wherein μ₁ is a first index whichindicates a first one of the plurality of frequency channels, wherein μ₂is a second index which indicates a second one of the plurality offrequency channels, wherein μ₃ is a third index which indicates a thirdone of the plurality of frequency channels, wherein H_(ρ)[μ₁, μ₂,μ₃]indicates the transfer function, wherein N_(DFT) ² indicates a square ofa number of the plurality of frequency channels of the frequency domain,wherein ϕNL,ρ is a number.

In an embodiment, the transfer function may, e.g., be normalized andnonlinear.

According to an embodiment, the analysis module 120 is configured todetermine the interference by applying a regular perturbation approach(e.g., Algorithm 1).

In an embodiment, the analysis module 120 is configured to determine theinterference by applying a regular logarithmic perturbation approach(e.g., Algorithm 2).

In an embodiment, the frequency domain may, e.g., be aregular-logarithmic frequency domain.

According to an embodiment, the transfer function may, e.g., depend on

${{H_{\nu}\left( e^{j\;\omega\; T} \right)} = {\frac{1}{T^{3}}{\sum\limits_{m \in {\mathbb{Z}}^{3}}{H_{\nu}\left( {\omega - \frac{2\pi m}{T}} \right)}}}}.$

In the following, embodiments of the present invention are described inmore detail.

At first, the notation and the overall system model is introduced.

The notation and basic definitions are now described.

Sets are denoted with calligraphic letters, e.g.,

is the set of data symbols, i.e., the symbol alphabet or signalconstellation. A set of numbers or finite fields are typeset inblackboard bold typeface, e.g., the set of real numbers is

. Bold letters, such as x, indicate vectors. If not stated otherwise, avector x=[x₁, x₂, . . . , x_(n)]^(T) of dimension n is a column vector,and the set of indices to the elements of the vector is

$\mathcal{I}\overset{def}{=}{\left\{ {1,\ldots\mspace{14mu},n} \right\}.}$

Non-bold italic letters, like x, are scalar variables, whereas non-boldRoman letters refer to constants, e.g., the imaginary number is j withj²=1. (·)^(T) denotes transposition and (·)^(H) is the Hermitiantransposition.

A real (bandpass) signal is typically described using the equivalentcomplex baseband (ECB) representation, i.e., we consider the complexenvelope x(t) ∈

with inphase (real) and quadrature (imaginary) component. The n-dimensional Fourier transform of a continuous-time signal x(t)=x(t₁,t₂, . . . , t_(n)) depending on the n-dimensional time vector t=[t₁, t₂,. . . , t_(n)]^(T) ∈R^(n) (in seconds) is denoted by X(ω)=

{x(t)}, and defined as [14, Ch. 4]

X ⁡ ( ω ) = ⁢ { x ⁡ ( t ) } ⁢ = def ⁢ ∫ ℝ n ⁢ x ⁡ ( t ) ⁢ e - j ⁢ ⁢ ω · t ⁢ d n ⁢ t( 1 ) x ⁡ ( t ) = - 1 ⁢ { X ⁡ ( ω ) } = 1 ( 2 ⁢ π ) n ⁢ ∫ ℝ n ⁢ X ⁡ ( ω ) ⁢ e j ⁢⁢ω · t ⁢ d n ⁢ ω . ( 2 )

Here, X(ω) is a continuous function of angular frequencies ω=[ω₁, ω₂, .. . , ω_(n)]^(T) ∈ R^(n) with ω=2πf and frequency f ∈R (in Hertz). Inthe exponential we made use of the dot product of vectors in R^(n) givenby ω·t=ω₁t₁+ω₂t₂+ . . . +ω_(n)t_(n). The integral is an n-fold multipleintegral over

^(n) and with integration boundaries at −∞ and ∞ in each dimension. Weuse the expression d^(n)t as shorthand for dt₁ dt₂ . . . dt_(n). For theone-dimensional case with n=1 the variable subscript is dropped. We mayalso write the correspondence as x(t) ∘-●X(ω) for short.

The n-dimensional discrete-time Fourier transform (DTFT) of adiscrete-time sequence <x[k]> with k=[k₁, k₂, . . . , k_(n)]^(T) ∈

^(n) with spacing T between symbols is periodic with 1/T in frequencydomain and denoted as X(e^(jωT))={circumflex over (F)}{x|k|}, anddefined as

X ⁡ ( e j ⁢ ω ⁢ T ) = ^ ⁢ { x ⁡ [ k ] } ⁢ = def ⁢ ∑ k ∈ ℤ n ⁢ x ⁡ [ k ] ⁢ e - j ⁢ ⁢ω · kT ( 3 ) x [ k ) = ^ - 1 ⁢ { X ⁡ ( e j ⁢ ω ⁢ T ) } = ( T 2 ⁢ π ) n ⁢ ∫ 𝕋 n⁢X ⁡ ( e j ⁢ ⁢ ω ⁢ ⁢ T ) ⁢ e j ⁢ ⁢ ω · kT ⁢ d n ⁢ ω . ( 4 )

The set of frequencies in the Nyquist interval is

${\mathbb{T}}\overset{def}{=}\left\{ {\omega \in {\mathbb{R}}} \middle| {{- \omega_{Nyq}} \leq \omega < \omega_{Nyq}} \right\}$

with the Nyquist (angular) frequency

$\omega_{Nyq}\overset{def}{=}{2{\pi/{\left( {2T} \right).}}}$

If a whole (finite-length) sequence is treated, this is indicated by thesquare bracket notation, i.e., <x[k]>

The notation Σ_(k∈z) _(n) is short for Σ_(k) ₁ ^(∞)=−∞ Σ_(k) ₂ ^(∞)=−∞.. . Σ_(k) _(n) ^(∞)=−∞.

Embodiments employ the so-called engineering notation of the Fouriertransform with a negative sign in the complex exponential (in theforward, i.e., time-to-frequency, direction) is used. This has immediateconsequences for the solution of the electro-magnetic wave equation (cf.Helmholtz equation), and therefore also for the NLSE. In the opticalcommunity, there exists no fixed convention with respect to the signnotation, e.g., some of the texts are written with the physicists'(e.g., [15, Eq. (2.2.8)] or [10]) and others with the engineering (e.g.,[16], [17, Eq. (A.4)]) notation in mind. Consequently, the derivationsshown here may differ marginally from some of the original sources.

Continuous-time signals are associated with meaningful physical units,e.g., the electrical field has typically units of volts per meter (V/m).The NLSE and the Manakov equation derived thereof are carried out inJones space over a quantity u(t)=[u_(x)(t), u_(y)(t)]^(T) ∈

² called the optical field envelope. The optical field envelope has thesame orientation as the associated electrical field but is renormalizeds.t. u^(H)u equals the instantaneous power given in watts (W). Here,signals are instead generally treated as dimensionless entities as thisconsiderably simplifies the notation when we move between the varioussignal domains (see, e.g., discussion in [18, P. 11] or [19, P. 230]).To this end, the nonlinearity coefficient y commonly given in W⁻¹m⁻¹ isalso renormalized to have units of m⁻¹, cf. II-B2.

To distinguish a two-dimensional complex vector u=[u_(x), u_(y)]^(T)

in Jones space from its associated three-dimensional real-valued vectorin Stokes space, we use decorated bold letters {right arrow over(u)}=[u₁, u₂, u₃]^(T) ∈

³. The (permuted) set of Pauli matrices is given by [20]

$\begin{matrix}{{\sigma_{1}\overset{def}{=}\begin{bmatrix}1 & 0 \\0 & {- 1}\end{bmatrix}}{\sigma_{2}\overset{def}{=}\begin{bmatrix}0 & 1 \\1 & 0\end{bmatrix}}{{\sigma_{3}\overset{def}{=}\begin{bmatrix}0 & {- j} \\j & 0\end{bmatrix}},}} & (5)\end{matrix}$

and the Pauli vector is

$\overset{->}{\sigma}\overset{def}{=}\left\lbrack {\sigma_{1},\sigma_{2},\sigma_{3}} \right\rbrack^{T}$

where each vector component is a 2×2 Pauli matrix. The relation betweenJones and Stokes space can then be established by the concise (symbolic)expression {right arrow over (u)}=u^(H){right arrow over (σ)}u to denotethe elementwise operation u_(i)=u^(H)σ_(o)u for all Stokes vectorcomponents i=1, 2, 3. The Stokes vector {right arrow over (u)} can alsobe expanded using the dot product with the Pauli vector to obtain thecomplex-valued 2×2 matrix with

$\begin{matrix}{\mspace{56mu}{{{\overset{->}{u} \cdot \overset{->}{\sigma}} = {{{u_{1}\sigma_{1}} + {u_{2}\sigma_{2}} + {u_{3}\sigma_{3}}} = \begin{bmatrix}{{u_{x}u_{x}^{*}} - {u_{y}u_{y}^{*}}} & {2u_{x}u_{y}^{*}} \\{2u_{x}^{*}u_{y}} & {{u_{y}u_{y}^{*}} - {u_{x}u_{x}^{*}}}\end{bmatrix}_{\text{?}}}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (6)\end{matrix}$

which will later be used to describe the instantaneous polarizationrotation around the Stokes vector {right arrow over (u)} using the Jonesformalism. We may also use the equality [20, Eq. (3.9)]

uu ^(H)=½(u^(H)u I+{right arrow over (u)}·{right arrow over (σ)})   (7)

with the identity matrix I and ∥u∥²=u^(H)u=u_(x)u_(x)*+u_(y)u_(y)*.

In the following, a system model according to embodiments is considered.

Some embodiments provide a point-to-point coherent optical transmissionover two planes of polarization in a single-mode fiber. This results ina complex-valued 2×2 multiple-input/multiple-output (MIMO) transmissionwhich is typically used for multiplexing. One of the major constraintsof today's fiber-optical transmission systems is the bandwidth ofelectronic devices which is orders of magnitude smaller than theavailable bandwidth of optical fibers. It is hence routine to usewavelength-division multiplexing (WDM), where a number of so-calledwavelength channels are transmitted simultaneously through the samefiber. Each wavelength signal is modulated on an individual laseroperated at different wavelengths such that the spectral support ofneighboring signals is not overlapping.

FIG. 2 illustrates a generic fiber-optical transmission system model. Inparticular, FIG. 2 shows the block diagram of a coherent opticaltransmission system exemplifying the digital, analog, and opticaldomains of a single wavelength channel. Within the bandwidth of awavelength channel, we can consider the optical end-to-end 2×2 MIMOchannel as frequency-flat if we neglect the effects of bandlimitingdevices (e.g., switching elements in a routed network). The nonlinearproperty of the fiber-optical transmission medium is the source ofinterference within and between different wavelength channels. In thefollowing, we will call the channel under consideration the probechannel, while a co-propagating wavelength channel is called interferer.This allows us to discriminate between self-channel interference (SCI)and cross-channel interference (XCI). In FIG. 2 the probe channel in theoptical domain is denoted by a subscript P, whereas interferers arelabeled by the channel index v with v∈{1, . . . , N_(ch)|v≠ρ}. Thevarious domains and its entities are discussed in the following.

FIG. 3a illustrates a transmitter frontend of a generic fiber-opticaltransmission system model.

FIG. 3b illustrates an optical channel of the generic fiber-opticaltransmission system model.

FIG. 3c illustrates an receiver frontend of the generic fiber-opticaltransmission system model and variables associated with the regularperturbation model.

In the following the transmitter frontend of FIG. 3a is described. Thetransmission system is fed with equiprobable source bits of the probe(and interferer) channel. The binary source generates uniform i.i.d.information bits q|K|∈F₂ at each discrete-time index K ∈Z. F₂ denotesthe Galois field of size two and Z is the set of integers. The binarysequence <q(K> is partitioned into binary tuples of length R_(m), s.t.q[k]=[q_(,)[k], . . . q_(R) _(m) [k]] ∈{0, 1}^(R) ^(m) , where k ∈Z isthe discrete-time index of the data symbols. Here, R_(m) is called therate of the modulation and will be equivalent to the number of bits pertransmitted data symbol, if we assume that the size of the symbol set isa power of two. Each R_(m)-tuple is associated with one of the possibledata symbols α=|a_(x), a_(y)|^(T)∈

C

² , i.e., with one of the constellation points. We say that the binaryR^(m)-tuples are mapped to the data symbols a ∈ A by a bijective mappingrule

: q

a.

The size of the data symbol set is M=|

=2^(R) ^(m) and we can write the alphabet as

$\mathcal{A}\overset{def}{=}{\left\{ {a_{1},\ldots,a_{M}} \right\} \Subset {{\mathbb{C}}^{2}.}}$

The symbol set has zero mean if not stated otherwise, that is E{a}=0,and we deliberately normalize the variance of the symbol set to

$\sigma_{a}^{2}\overset{def}{=}{{E\left\{ {a}^{2} \right\}} = 1}$

(the expectation is denoted by E{·} and the Euclidean vector norm is∥·∥. For reasons of readability we denote the data symbols of theinterfering channels by b_(v)[k]·.

The discrete-time data symbols a[k] are converted to the continuous-timetransmit signal 8(t) by means of pulse-shaping constituting thedigital-to-analog (D/A) transition, cf. FIG. 3a . We can express thetransmit signal 8(t)=[s₁(t), s₂(t), s₂(t)]_(T) ∈

² as a function of the data symbols with

$\begin{matrix}{{{s(t)} = {T \cdot {\sum\limits_{k \in {\mathbb{Z}}}{{a\lbrack k\rbrack}{h_{T}\left( {t - {kT}} \right)}}}}},} & (8)\end{matrix}$

where 8(t) is a superposition of a time-shifted (with symbol period T)basic pulses h_(T)(t) weighted by the data symbols. The pre-factor T isused to preserve a dimensionless signal in the continuous-time domain(cf. [18, P. 11]). We assume that the transmit pulse has √{square rootover (Nyquist)} property, i.e., |H_(T)(ω)|² has Nyquist property withthe Fourier pair h_(T)(t) ∘-●H_(T)(ω). To keep the following derivationstractable, all wavelength channels transmit at the same symbol rate

$R_{s}\overset{def}{=}{1/T}$

as me prooe cnannei. ne pulse energy E_(T) of the probe channel is givenby [18, Eq. (2.2.22)]

$\begin{matrix}{E_{T} = {{\int_{- \infty}^{\infty}{{{T \cdot {h_{T}(t)}}}^{2}{dt}}} = {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{{T \cdot {H_{T}(\omega)}}}^{2}{{d\omega}.}}}}}} & (9)\end{matrix}$

The pulse energy E_(T) has the unit seconds due to the normalization ofthe signals. Using the symbol energy

$E_{s}\overset{def}{=}{\sigma_{a}^{2}{E_{T}.}}$

the average signal power P calculates to [18, Eq. (4.1.1)]

$\begin{matrix}{P\overset{def}{=}{{\frac{1}{T}{\int_{0}^{T}{E\left\{ {{s(t)}}^{2} \right\}{dt}}}} = {{\frac{\sigma_{a}^{2}}{T}E_{T}} = {\frac{E_{s}}{T}.}}}} & (10)\end{matrix}$

Since, see above, the variance of the data symbols σ_(a) ² is fixed to1, the transmit power P is directly adjusted via the pulse energy E_(T).The corresponding quantities related to one of the interferers areindicated by the subscript V·.

In the following, an optical channel according to FIG. 3b is described.The electrical-to-optical (E/O) conversion is performed by an idealdual-polarization (DP) inphase-quadrature (IQ) converter. The twoelements of the transmit signal 8_(v)(t) correspond to the modulatedoptical signals in the x- and y-polarization. The optical field envelopeu_(v)(z, t) of each wavelength channel

u _(v)(0, t)=s _(v)(t) exp (jΔω _(v)t),   (11)

is modulated at its angular carrier frequency ω_(v)=ω₀+Δω_(v) at theinput of the optical transmission line z=0. Here, ω₀=2πf₀ is the centerfrequency of the signaling regime of interest. For the probe channel,the carrier frequency ω_(σ) is to coincide with ω_(O) such that Δω_(σ)=0and u_(σ)(0, t)=8_(σ)(t). The transmitter frontend of the probe channelis shown in FIG. 3a .

The N_(ch) wavelength signals u_(v)(0, t) at z=0 are combined by anideal optical multiplexer to a single WDM signal, cf. FIG. 3b . Theoptical field envelope before transmission is then

$\begin{matrix}{{u\left( {0,t} \right)} = {{\sum\limits_{v = 1}^{N_{ch}}{u_{v}\left( {0,t} \right)}} = {\sum\limits_{v = 1}^{N_{ch}}{{s_{v}(t)}{\exp\left( {{j{\Delta\omega}}_{v}t} \right)}}}}} & (12) \\ & \; \\{{{U\left( {0,\omega} \right)} = {{\sum\limits_{v = 1}^{N_{ch}}{U_{v}\left( {0,\omega} \right)}} = {\sum\limits_{v = A}^{N_{ch}}{S_{v}\left( {\omega - {\Delta\omega}_{v}} \right)}}}},} & (13)\end{matrix}$

with the Fourier pairs 8_(v)(t)

S_(v)(ω) and u(0, t)

U (0, ω). Any initial phase and laser phase noise (PN) are neglected tofocus only on deterministic distortions. The optical field envelope isthe ECB representation of the optical field u_(o)(z, t) in the passbandnotation

$\begin{matrix}{{{u_{o}\left( {z,t} \right)}\overset{def}{=}{{u\left( {z,t} \right)} \cdot {\exp\left( {{{j\omega}_{0}t} - {{{j\beta}_{0}(z)}z}} \right)}}},} & (14)\end{matrix}$

which is known as the slowly varying amplitude approximation [15, Eq.(2.4.5)]. For consistency of notation we treat the optical fieldenvelope as a dimensionless entity (in accordance with the electricalsignals). The optical field propagates in Z-direction (the dimension Zhas units of meter) with the local propagation constant β₀(z)=β(z, ω₀),β(z, ω) is the space and frequency-dependent propagation constant. ATaylor expansion of β(z, ω) is performed around ω₀ with the derivativesof β(z, ω) represented by the coefficients [15, Eq. (2.4.4)]

$\begin{matrix}{{{\beta_{n}(z)}\overset{def}{=}\left. \frac{\partial^{n}{\beta\left( {z,\omega} \right)}}{\partial\omega^{n}} \right|_{\omega = \omega_{0}}},{n \in {{\mathbb{N}}.}}} & (15)\end{matrix}$

Here, we only consider coefficients up to second order, i.e., n ∈{0,1,2}. We also introduce the path-average4 dispersion length

$\begin{matrix}{{L_{D}\overset{def}{=}\frac{1}{2\pi{\beta_{2}}R_{s}^{2}}},} & (16)\end{matrix}$

which denotes the distance after which two spectral components spacedB=R_(s) Hertz apart, experience a differential group delay of T=1/R_(s)due to chromatic dispersion (CD). We can equivalently define thewalk-off length of the probe and one interfering wavelength channel as

$\begin{matrix}{{L_{{wo},v}\overset{def}{=}\frac{1}{{{{\Delta\omega}_{v}\beta_{2}}}R_{s}}},} & (17)\end{matrix}$

which quantifies the fiber length that must be propagated in order forthe v^(th) wavelength channel to walk off by one symbol from the probechannel.

4We discriminate between local (i.e., α(z), β(z), γ(z)) and path-average(i.e., α, β, γ) properties of the transmission link. The latter areimplicitly indicated if the z-argument of the local property is omitted,e.g.,

$\beta_{2}\overset{\Delta}{=}{\frac{1}{L}{\int_{0}^{L}{{\beta_{2}(\zeta)}d\;{\zeta.}}}}$

Now, signal propagation is considered.

In the absence of noise, the two dominating effects governing thepropagation of the optical signal in the fiber are dispersion—expressedby the z-profile of the fiber dispersion coefficient β₂(z)—and nonlinearsignal-signal interactions. Generation of the so-termed local NLIdepends jointly on the local fiber nonlinearity coefficient γ(z) and thez-profile of the optical signal power. For ease of the derivation, weassume that all z-dependent variation in γ(z) can be equivalentlyexpressed in a variation of either a local gain g(z) or the local fiberattenuation α(z). We also neglect the time- (and frequency-) dependencyof the attenuation, gain, and nonlinearity coefficient.

The interplay between the optical signal, dispersion, and nonlinearinteraction is all combined in the noiseless Manakov equation. It is acoupled set of partial differential equations in time-domain for theoptical field envelope u(z, t) in the ECB, and the derivative is takenw.r.t. propagation distance z ∈R and to the retarded time t ∈R. Theretarded time is defined as

${t\overset{def}{=}{t^{\prime} - {z/v_{g}}}},$

where t′ is me pnysical time and v_(g) is the (path-average) groupvelocity v_(g)=1/β₁ of the probe channel [15, Eq. (2.4.8)]. It can beunderstood as a time frame that moves at the same average velocity asthe probe to cancel out any group delay at the reference frequencyω_(σ)=ω₀. All other frequencies experience a residual group delayrelative to the reference frequency due to CD.

The propagation of u(z, t) in the signaling regime of interest isgoverned by [17, Eq.(6.26)]

$\begin{matrix}{{\frac{\partial}{\partial z}u} = {{j\frac{\beta_{2}(z)}{2}\frac{\partial^{2}}{\partial z^{2}}u} + {\frac{{g(z)} - {\alpha(z)}}{2}u} - {j{\gamma(z)}\frac{8}{9}{u}^{2}{u.}}}} & (18)\end{matrix}$

The space- and time-dependency of u(z, t) is omitted here for compactnotation. By allowing the local gain coefficient g(z) to contain Diracδ-functions one can capture the z-dependence of an amplification scheme,i.e., based on lumped erbium-doped fiber amplifier (EDFA) or Ramanamplification. Polarization-dependent effects such as birefringence andpolarization mode dispersion (PMD) are neglected limiting the followingderivations to the practically relevant case of low-PMD fibers. We alsoassume that all wavelength channels are co-polarized, i.e., modulated onpolarization axes parallel to the ones of the probe channel.

Now, the dispersion profile is considered.

The accumulated dispersion is a function that satisfies [21, Eq. (8)]

$\begin{matrix}{{\frac{d\;{\mathcal{B}(z)}}{dz} = {\beta_{2}(z)}}.} & (19)\end{matrix}$

Here,

(z) can be used to express a z-dependency in the dispersion profile,i.e., lumped dispersion compensation by inline dispersion compensationor simply a transmission link with distinct fiber properties acrossmultiple spans. We obtain

(z)=∫₀ ^(z)β₂(ζ)dζ+

₀,   (20)

where

$\mathcal{B}_{0}\overset{def}{=}{\mathcal{B}(0)}$

is the amount of pre-dispersion (in units of squared seconds, typicallygiven in ps²) at the beginning of the transmission line.

Now, the power profile is considered.

To describe the power evolution of u(z, t). we introduce the normalizedpower profile

(z) as a function that satisfies the equation [21, Eq. (7)]

$\begin{matrix}{{\frac{d\;(z)}{dz} = {\left( {{g(z)} - {\alpha(z)}} \right)(z)}},} & (21)\end{matrix}$

with boundary condition P(0)=P(1,)=1, i.e., the last optical amplifierresets the signal power to the transmit power.

The Z -dependence on α(z) allows for varying attenuation coefficientsover different spans. In writing (21) we assumed that both the localgain coefficient and attenuation coefficient are frequency-independent.We may also define the logarithmic gain/loss profile as

$\begin{matrix}{{(z)}\overset{def}{=}{{\ln\left( {\mathcal{P}(z)} \right)} = {\int_{0}^{z}{\left( {{g(\zeta)} - {\alpha(\zeta)}} \right)d\;{\zeta.}}}}} & (22)\end{matrix}$

The last expression in (22) is obtained by solving (21) for P(z)=

The boundary conditions on P(z) immediately give the boundary condition

(0)=

(L)=0.

We can now define the impulse response and transfer function of thelinear channel—that is, when the fiber nonlinearity coefficient is zero,i.e., γ=0 in (18). To that end, we define the optical field envelopeu_(LIN)(z, t)

U_(LIN)(z, ω) that propagates solely according to linear effects withthe boundary condition u_(LIN)(0, t)=u(0, t) at the input of thetransmission link. The linear channel transfer function and impulseresponse is then given by

$\begin{matrix}{{H_{C}\left( {z,\omega} \right)}\overset{def}{=}{\exp\left( \frac{{(z)} - {j\omega^{2}{\mathcal{B}(z)}}}{2} \right)}} & (23) \\{{{h_{C}\left( {z,t} \right)} = {\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{j\;{\mathcal{B}(z)}}}{\exp\left( \frac{{(z)} + {j{t^{2}/{\mathcal{B}(z)}}}}{2} \right)}}},} & (24)\end{matrix}$

which represents the joint effect of chromatic dispersion and thegain/loss variation along the link. We finally have the linear channelrelation in time-domain u_(LIN)(z) t)=h_(C)(z, t)* u_(LIN)(0 ,t) andfrequency-domain U_(LIN)(z, ω)=H_(C)(z, ω)U_(LIN) (0, ω). which will beused in the following to derive the first-order perturbation method.

In the following, a receiver frontend according to FIG. 3c is described.Again, we assume ideal optical-to-electrical (O/E) and analog-to-digital(A/D) conversion. The received continuous-time, optical signal u(L, t)is first matched filtered w.r.t. the linear channel response andtransmit pulse and then sampled at the symbol period T, cf. FIG. 3 (c).The receiver frontend hence also compensates for any residual link lossand performs perfect CD compensation. Note, that the analog frontend isusually realized using an oversampled digital representation. E.g., CDcompensation is typically performed in the (oversampled) digital domain.Here, we favour to conceptually incorporate it in the analog domainsince it significantly simplifies notation in the derivation of theend-to-end channel model. The transfer function of the entire cascade ofthe receiver frontend is given by

$\begin{matrix}{{H_{R}(\omega)} = {\frac{T}{E_{T}}{H_{C}^{*}\left( {L,\omega} \right)}{{H_{T}^{*}(\omega)}.}}} & (25)\end{matrix}$

The factor T/E_(T) re-normalizes the received signal to the variance ofthe constellation σ_(a) ². Since we only consider T-spaced sampling anyfractional sampling phase-offset or timing synchronization is alreadyincorporated as suited delay in the receive filter h_(R)(t). s.t. thetransmitted and received sequence of the probe are perfectly aligned intime.

Note, that the time delay L/v_(g) at ω₀ and any initial phase β₀ hasalready been canceled from the propagation equation.

In the following, first-order perturbation is considered.

A concept of fiber-optical channel models based on the perturbationmethod is to assume that nonlinear distortions are weak compared to itssource, i.e., the linearly propagating signal. Starting from thispremise the regular perturbation (RP) approach for the opticalend-to-end channel is written as

u(L, t)=u _(LIN)(L, t) Δu(L, t),   (26)

where u_(LIN)(z, t) ∈

² is the signal propagating according to the linear effects, i.e.,according to (23), (24). In this context, the nonlinear distortion Δu(z,t) ∈

² is termed perturbation, which is generated locally according tononlinear signal-signal interaction and is then propagated linearly andindependently of the signal u^(LIN)(z, t) to the end of the opticalchannel at z=L. We assume that the optical perturbation at z=0 is zero,i.e., Δu(0, t)=0. The received signal is then given as the sum of thesolution for the linearly propagating signal and the accumulatedperturbation representing the accumulated nonlinear effects. Anobjective here is to develop the input/output relation of the equivalentdiscrete-time end-to-end channel in the form of

y[k]=a[k]+Δa[k],   (27)

where the total NLI is absorbed into a single discrete-time perturbativeterm Δa[k], cf. FIG. 3 (c). To that end, we start with a known RPsolution of the optical end-to-end relation and successively incorporatethe used components according to FIG. 2 and FIG. 3.

Now, the optical end-to-end channel is considered.

The solution to the optical perturbation after transmission at z=L isgiven in frequency-domain by [4, Eq. (12)], [22, Eq. (2)], [23, Eq.(4)], [24, Eq. (24)-(27)],

$\begin{matrix}{{{\Delta\;{U\left( {L,\omega} \right)}} = {{- j}\;\gamma\frac{8}{9}\frac{L_{eff}}{\left( {2\pi} \right)^{2}}{H_{C}\left( {L,\omega} \right)} \times {\int_{{\mathbb{R}}^{2}}{{\underset{¯}{U}\left( {\omega,v_{1},v_{2}} \right)}{H_{NL}\left( {v_{1},v_{2}} \right)}d^{2}v}}}},} & (28)\end{matrix}$

with the normalized nonlinear transfer function H_(NL)(V₁, v₂) and

${{\underset{\_}{U}\left( {\omega,v_{1},v_{2}} \right)}\overset{def}{=}{{U\left( {0,{\omega + v_{2}}} \right)}{U^{H}\left( {0,{\omega + v_{1} + v_{2}}} \right)}{U\left( {0,{\omega + v_{1}}} \right)}}},$

i.e., a term that depends on the optical field envelope at the input ofthe transmission system. Note, that we made use of the common variablesubstitution

$\begin{matrix}{\omega_{1}\overset{def}{=}{\omega\; + v_{1}}} & (29) \\{\omega_{2}\overset{def}{=}{\omega + v_{1} + v_{2}}} & (30) \\{{\omega_{3}\overset{def}{=}{{\omega - \omega_{1} + \omega_{2}} = {\omega + v_{2}}}},} & (31)\end{matrix}$

to express the field U in terms of difference frequencies υ¹ and υ²relative to ω. FIG. 4a and FIG. 4b summarizes definitions of the time-and frequency variables that are used throughout this text. The integralover

² in (28) can also be performed w.r.t. ω₁ and ω₂

FIG. 4a illustrates definitions of variables in the time-domain. FIG. 4billustrates definitions of variables in the frequency-domain. Both T₁,T₂and υ₁, υ₂ can take positive and negative values in R.

Equation (28) shows that the first-order RP method can be understood asa FWM process with un-depleted pumps where three wavelengths affect afourth. Equivalently, one can think of the joint annihilation andcreation of two two-photon pairs (i.e., with four frequencies involved)preserving both energy (frequency matching) and momentum (phasematching) during the interaction [25, FIG. 7.2.5]. The conjugate fieldcorresponds to the inverse process where photon creation andannihilation is interchanged.

FIG. 5 illustrates a magnitude in logarithmic scale of a single-spannonlinear transfer function for β₂=−21 ps²/km,

₀=0 ps², 10 log₁₀ e^(α)=0.2 dB/km and L_(sp)=100 km over the differencefrequencies υ₁ and υ₂ normalized to R_(s)=64 GBd. The red line denotesH_(NL)(ξ) which only depends on the scalar ξ=υ₁υ₂. (Part for υ₁>υ₂ notshown).

The normalized nonlinear transfer function is a measure of the phasematching condition and defined as

$\begin{matrix}\begin{matrix}{{H_{NL}\left( {v_{1},v_{2}} \right)}\overset{def}{=}{\frac{1}{L_{eff}}{\int_{0}^{L}{{\exp\left( {{(\zeta)} + {jv_{1}v_{2}{\mathcal{B}(\zeta)}}} \right)}d\;\zeta}}}} \\{= {\frac{1}{L_{eff}}{\int_{0}^{L}{{H_{C}^{*}\left( {\zeta,\sqrt{v_{1}v_{2}}} \right)}^{2}d\;{\zeta.}}}}}\end{matrix} & (34)\end{matrix}$

The pre-factor is the effective length of the whole transmission linkdefined as

$\begin{matrix}{{L_{eff}\overset{def}{=}{{\int_{0}^{L}{(\zeta)d\zeta}} = {\int_{0}^{L}{{\exp\left( {(\zeta)} \right)}d\;\zeta}}}},} & (35)\end{matrix}$

and acts as a normalization constant s.t. H_(NL)(0, 0)=1.

The phase mismatch Δβ. i.e., the difference in the (path-average)propagation constant due to dispersion, is defined as [15, Eq. (6.3.19)]

$\begin{matrix}\begin{matrix}{{\Delta\beta}\overset{def}{=}{{\beta\;(\omega)} - {\beta\left( \omega_{1} \right)} + {\beta\left( \omega_{2} \right)} - {\beta\left( \omega_{3} \right)}}} \\{= {\frac{\beta_{2}}{2}\left( {\omega^{2} - \omega_{1}^{2} + \omega_{2}^{2} - \left( {\omega - \omega_{1} + \omega_{2}} \right)^{2}} \right)}} \\{{= {{{\beta_{2}\left( {\omega_{1} - \omega} \right)}\left( {\omega_{2} - \omega_{1}} \right)} = {\beta_{2}v_{1}v_{2}}}},}\end{matrix} & (36)\end{matrix}$

where the propagation constants at the four frequencies are developed ina second-order Taylor series according to (15). E.g., for transmissionsystems without inline dispersion compensation and zero pre-dispersion

₀=0, we have

(z)=β₂z and the phase mismatch Δβ can be found in the argument of theexponential in (34) with v₁v₂

(z)=Δβz.

In the context of the equivalent approach following the regular VSTF[3], [4], [24], the nonlinear transfer function H_(NL)(v₁, v₂) is alsoreferred to as 3rd-order Volterra kernel. Closed form analyticalsolutions to (34) can be obtained for single-span or homogeneousmulti-span systems [24], [26]. It is noteworthy, that H_(NL)(v₁, v₂)contains all information about the transmission link characterized bythe dispersion profile (including CD pre-compensation

₀, cf. (20)) and the gain/loss profile.

FIG. 5 shows the magnitude of H_(NL)(v₁, v₂) exemplifying a single-spanstandard single-mode fiber (SSMF) link. Note, that H_(NL)(v₁, v₂)depends in fact on the product

$\xi\overset{def}{=}{v_{1}v_{2}}$

and is hence a hyperbolic function in two dimensions [27, Sec. VIII](cf. the contour in FIG. 5). The bold red line drawn into the diagonalcross section in FIG. 5 is the corresponding nonlinear transfer functionH_(NL)(ξ) which only depends on the scalar variable

$\xi\overset{def}{=}{v_{1}{v_{2}.}}$

FIG. 6 illustrates a magnitude in logarithmic scale of a single-spannonlinear transfer function for β₂=−21 ps²/km,

₀=0 ps², 10 log₁₀ e^(α)=0.2 dB/km and L_(sp8)=100 km over ξ=v₁v₂. Thenormalization by (2πR_(s))² relates H_(NL)(ξ) to the probe's spectralwidth. The width of |H_(NL)(ξ/R_(s) ²)|² is then proportional to 1/

_(T,σ)=L_(D)/L_(eff) ∝ R_(s) ⁻², i.e., doubling R_(s) reduces thespectral width by a factor of 4.

FIG. 6 shows the H_(NL)(ξ) over the normalized variable ξ/2πR_(s))² torelate the nonlinear transfer function to the spectral width of theprobe channel. The spectral width of |H_(NL)(ξ/(2πR_(s))²)|² isproportional to the inverse dimensionless map strength

1 / T , ρ ⁢ = def ⁢ L D / L eff

closely related to the nonlinear diffusion bandwidth defined in [22].Conversely, the map strength

_(T,σ) quantifies the number of nonlinearly interacting pulses in timeover the effective length L_(eff) within the probe channel [28]. It istherefore a direct measure of intra-channel (i.e., SCI) nonlineareffects [29]. The relevant quantity for inter-channel (i.e., XCI)effects is given by

$S_{T,v}\overset{def}{=}{L_{eff}/L_{{wo},v}}$

(with v≠σ) where the temporal walk-off between wavelength channels isthe relevant length scale. In [23] it was shown that H_(NL)(ξ) isrelated to the power-weighted dispersion distribution (PWDD) by a(one-dimensional) Fourier transformation (w.r.t. the scalar variable ξ)and has a time-domain counterpart which is discussed in the nextparagraph.

In the following, the electrical end-to-end channel is considered.

To derive the discrete-time end-to-end channel model the filter cascadeof the linear receiver frontend is subsequently applied to ΔU(L, ω). Theperturbation ΔS (ω) (i.e., the perturbation in the electrical domainfollowing our terminology, cf. FIG. 3c ) is obtained by

ΔS(ω)=H _(C)*(L, ω)ΔU(L, ω),   (39)

which cancels out the leading term H_(C)(L, ω) in (28) since |H_(C)(L,ω)|=1. The result is shown in (32) at the bottom of this page.Remarkably, there exists an equivalent time-domain representation Δs(t)

ΔS(ω) shown in (33) where the Fourier relation is derived in Appendix A.The time-domain perturbation Δs(t) has the same form as itsfrequency-domain counterpart, i.e., the integrand is constituted by therespective time-domain representation of the optical signal and thedouble integral is performed over the time variables T₁ and T₂ (cf.FIGS. 4 (a) and [23], [30]).

The frequency matching with

$\omega_{3}\overset{def}{=}{\omega - \omega_{1} + \omega_{2}}$

is translated to a temporal matching 8

$t_{3}\overset{def}{=}{t - t_{1} + t_{2}}$

(cf. [31 ]), i.e., the selection rules of FWM apply both in time andfrequency. The temporal matching is not to be confused with the phasematching condition in (34), (36).

Remarkably, the time-domain kernel h_(NL)(T₁, T₂) is related toh_(NL)(υ₁, υ₂) by an inverse two-dimensional (2D) Fourier transform (cf.[30, Appx.] and [28, Eq. (6)]) which can be written as

h N ⁢ L ⁡ ( τ 1 , τ 2 ) = ⁢ h N ⁢ L ⁡ ( τ ) = - 1 ⁢ { H N ⁢ L ⁡ ( v ) } = ⁢ 1 Leff ⁢ ∫ 0 L ⁢ 1 2 ⁢ π ⁢  ℬ ⁡ ( ζ )  ⁢ exp ⁡ ( ⁢ ( ζ ) - j ⁢ τ 1 ⁢ τ 2 ℬ ⁡ ( ζ ) )⁢d ⁢ ⁢ ζ = ℬ ⁡ ( z ) ≤ 0 ⁢ ⁢ j L eff ⁢ ∫ 0 L ⁢ h C * ⁡ ( ζ , τ 1 ⁢ τ 2 ) 2 ⁢ d ⁢ ⁢ ζ, ( 40 )

with the tuples T=[T₁, T₂]^(T) and υ=[υ₁υ₂]^(T). The time-domain kernelmaintains its hyperbolic form as it is a function of the product T₁T₂.Also note the duality to (34), where in both representations thenonlinear transfer function can be understood as the path-average (cf.[32]) over an expression related to the linear channel responseh_(C)(z,t)

H_(C)(z,ω). Note, that in (40) the condition on β(z)≤0 (which istypically fulfilled in the anomalous dispersion regime with β₂<0) isused to obtain the simple result without cumbersome differentiation ofthe term |β(z)|.

The next step is to resolve the perturbation Δs (t)

ΔS(ω) into contributions originating from SCI, XCI or multichannelinterference (MCI). We notice from FIG. 6 that, given R_(s) issufficiently large, |H_(NL)(ξ)|² vanishes if ϵ»(2πR_(s)), i.e., if thephase matching condition is not properly met. Conversely, if thespectral width of |N_(NL)(ξ/R_(s) ²)|² (or equivalently the inverse mapstrength 1/δ_(T,ρ)) is small enough, the integrand in (32), (33) can befactored into a SCI and XCI term, i.e., mixing terms that originateeither from within the probe channel (both υ₁<2πR_(s) and υ₂<2πR_(s)) orfrom within the probe channel and a single interfering wavelengthchannel (either υ₁<2πR_(s), or υ₂<2πR_(s)). Mixing terms originatingfrom MCI are only relevant for small R_(s). We hence neglect any FWMterms involving more than two wavelength channels.

The optical field envelope u(0, t)

U(0, ω) in (32), (33) is now expanded according to (12), (13). Bydefinition we have Δω_(ρ)=0 and we can expand the triple product of U(0,ω) in (32) as

$\begin{matrix}{{{UU}^{H}U} = {\underset{\underset{SCI}{︸}}{U_{\rho}U_{\rho}^{H}U_{\rho}} + {\sum\limits_{v \neq \rho}\underset{\underset{XCI}{︸}}{\left( {{U_{v}U_{v}^{H}U_{\rho}} + {U_{\rho}U_{v}^{H}U_{v}}} \right)}}}} & (41)\end{matrix}$

where the frequency-dependency of U(0, ω) is omitted for short notation.The XCI term has two contributionsthe first results from an interactionwhere ω₃ and ω₂ are from the v^(th) interfering wavelength channel and ωand ω₁ are within the probe's support (υ₂→Δω_(v) in FIG. 4 (b)). Thesecond involves an interaction where ω₂ and ω₁ are from the interferingwavelength channel and ω and ω₃ are from the probe channel (υ₁→Δω_(v)).

We can exploit the symmetry of the nonlinear transfer functionH_(NL)(υ₁, υ₂)=H_(NL)(υ₂, υ₁) to simplify the XCI expression in (41).Since U_(v) ^(H)U_(v) is a scalar, we have U_(ρ)U_(v) ^(H)U_(v)=U_(v)^(H)U_(v)U_(ρ). The 2×2 identity matrix I is used to factor the XCIexpression in a v- and ρ-dependent term. We obtain with the definitionof the electrical signal of each wavelength channel (cf. (12), (13))after rearranging some terms

$\begin{matrix}{{{{U\left( {0,\omega_{3}} \right)}{U^{H}\left( {0,\omega_{2}} \right)}{U\left( {0,\omega_{1}} \right)}{H_{NL}\left( {{\omega_{2} - \omega_{3}},{\omega_{2} - \omega_{1}}} \right)}} = {{{S_{\rho}\left( \omega_{1} \right)}{S_{\rho}^{H}\left( \omega_{2} \right)}{S_{\rho}\left( \omega_{3} \right)}{H_{NL}\left( {{\omega_{2} - \omega_{1}},{\omega_{2} - \omega_{3}}} \right)}} + {\sum\limits_{v \neq \rho}{\left( {{{S_{v}\left( \omega_{1} \right)}{S_{v}^{H}\left( \omega_{2} \right)}} + {{S_{v}^{H}\left( \omega_{2} \right)}{S_{v}\left( \omega_{1} \right)}I}} \right){S_{\rho}\left( \omega_{3} \right)} \times {H_{NL}\left( {\underset{\underset{v_{2}}{︸}}{\omega_{2} - \omega_{1}},{\underset{\underset{v_{1}}{︸}}{\omega_{2} - \omega_{3}} - {\Delta\; w_{v}}}} \right)}}}}},} & (42)\end{matrix}$

which now corresponds to the case that ω₃ always lays in the support ofthe probe 10. The signals of the interfering wavelength channels are nowrepresented in their respective ECB and the relative frequency offsetΔω_(v) is accounted for in the modified nonlinear transfer functionH_(NL).

At this point, considering (32) and (42), we formulated the relationbetween the perturbation at the probe ΔS(ω) after chromatic dispersioncompensation and the transmit spectra S_(v)(ω) of the probe and theinterferers in their respective baseband. The remaining operation in thereceiver cascade is to perform matched filtering w.r.t. the transmitpulse and then to perform T-spaced sampling. An alternative formulationwith ω1 in the support of the probe is obtained by exchanging thesubscripts of ω1 and ω3 in frequency-domain and t1 and t3 intime-domain.

Now, the discrete-time end-to-end channel is considered.

We recap that the periodic spectrum X(e^(jωT)) of the sampled signal

${x\lbrack k\rbrack}\overset{def}{=}{x({kT})}$

is related to the aliased spectrum of the continuous-time signal x(t)over the Nyquist interval

by

$\begin{matrix}{{X\left( e^{j\;\omega\; T} \right)}\overset{def}{=}{{{ALIAS}\left\{ {X(\omega)} \right\}} = {\frac{1}{T}{\sum\limits_{m \in {\mathbb{Z}}}{{X\left( {\omega - \frac{2\pi m}{T}} \right)}.}}}}} & (43)\end{matrix}$

The matched filter H_(T)*(ω) and the aliasing operator are used totranslate (32), (33) to the equivalent discrete-time form in (37), (38)exemplarily for the SCI contribution Δa^(SCI). The total perturbationinflicted on the probe channel is Δa[k]=Δa^(SCI)[k]+Δa^(SCI)[k]. In(37), (38) we use the 1/T-periodic spectrum A(e^(jωT)) which is relatedto the discrete-time sequence

a[k]

by a DTFT A(e^(jωT))=

{a[k]}. The channel-dependent nonlinear length is

$L_{{NL},\nu}\overset{def}{=}{1/\left( {\gamma P_{\nu}} \right)}$

and P_(v) is the optical launch power of the v^(th) wavelength channel.The normalized nonlinear end-to-end transfer function H_(v)(ω)=H_(v)(ω₁,ω₂, ω₃) characterizes the nonlinear cross-talk from the v^(th)wavelength channel to the probe channel. In particular, H_(ρ)(ω)describes SCI and H_(v)(ω) with v≠ρ describes XCI. It is defined as

$\begin{matrix}{{{H_{v}(\omega)}\overset{def}{=}{{T \cdot {H_{T,v}\left( \omega_{1} \right)}}{T \cdot {{H_{T,v}^{*}\left( \omega_{2} \right)}/P_{v}}} \times {T \cdot {H_{T,\rho}\left( \omega_{3} \right)}}{T \cdot {{H_{T,\rho}^{*}\left( {\omega_{1} - \omega_{2} + \omega_{3}} \right)}/E_{T^{\prime}}}} \times {H_{NL}\left( {{\omega_{2} - \omega_{1}},\ {\omega_{2} - \omega_{3} - {\Delta\;\omega_{v}}}} \right)}}},} & (44)\end{matrix}$

and its periodic continuation, i.e., the aliased discrete-timeequivalent is given by

$\begin{matrix}{{{H_{v}\left( e^{j\;\omega\; T} \right)} = {\frac{1}{T^{3}}{\sum\limits_{m \in {\mathbb{Z}}^{3}}{H_{\nu}\left( {\omega - \frac{2\pi m}{T}} \right)}}}},} & (45)\end{matrix}$

where the three-fold aliasing is done along each frequency dimensionwith ω=[ω₁, ω₂, ω₃]^(T) and m=[m₁, m₂, m₃]^(T). The normalization in(44) is done s.t. H_(ρ)(e^(j0T)=)1 and dimensionless. Note, that bydefinition the optical launch power P_(v) of the v^(th) wavelengthchannel is related to the pulse energy of H_(T,v)(ω) in (9), (10).

The nonlinear end-to-end transfer function in (44) depends on thecharacteristics of the transmission link, comprised by H_(NL)(·, ·), thecharacteristics of the pulse-shapes of the probe and interferingwavelength channel (assuming matched filtering w.r.t. the channel andthe probe's transmit pulse) and the frequency offset Δω_(v) betweenprobe and interferer.

It is remarkable that the integration in (37) is over the twofold tuple[ω₁, ω₂]^(T) while the time-domain summation in (38) is over threeindependent variables k=[k₁, k₂, k₃]^(T) ∈

³. This is a consequence of the time-frequency relation betweenconvolution and element-wise multiplication. The temporal matching usedfor the optical field in (33) is now canceled in (38) due to theconvolution with the matched filter h_(T)*(−T), i.e., k₃ does not dependon k₁ and k₂ unlike

$t_{3}\overset{def}{=}{t - t_{1} + {t_{2}.}}$

Note, that the frequency variable ω₃ in (37) still complies with thefrequency matching ω₃=ω−ω₁+ω₂ but may be outside the Nyquist interval

. Due to the 1/T-periodicity of the spectrum A(e^(jωT)) any frequencycomponent outside

, is effectively folded back into the Nyquist interval by addition ofinteger multiples of ω^(Nyq) (denoted by the FOLD{·} operation in (37)).

The XCI complement to (37) reads

$\begin{matrix}{{\Delta\;{A^{XCI}\left( e^{j\;\omega\; T} \right)}} = {{- j}{\sum\limits_{\nu \neq p}{\frac{8}{9}\frac{L_{eff}}{L_{{NL},\nu}}\frac{T^{2}}{\left( {2\pi} \right)^{2}}{\int_{{\mathbb{T}}^{2}}{\times \left( {{{B_{\nu}\left( e^{j\omega_{1}T} \right)}{B_{\nu}^{H}\left( e^{j\;\omega_{2}T} \right)}} + {{B_{\nu}^{H}\left( e^{j\;\omega_{2}T} \right)}{B_{\nu}\left( e^{j\;\omega_{1}T} \right)}I}} \right) \times {A\left( e^{j\omega_{3}T} \right)}{H_{\nu}\left( e^{j\;\omega\; T} \right)}d^{2}{\omega.}}}}}}} & (46)\end{matrix}$

The time-domain description of the T-spaced channel model in (38) isequivalent to the pulse-collision picture (cf. [13, Eq. (3-4)] and [33,Eq. (3-4)]) and the XCI result is repeated here for completeness

$\begin{matrix}{{\Delta{a^{XCI}\lbrack k\rbrack}} = {{- j}{\sum\limits_{v \neq \rho}{\frac{8}{9}\frac{L_{eff}}{L_{{NL},\nu}}{\sum\limits_{\kappa \in {\mathbb{Z}}^{3}}{\left( {{{b_{\nu}\left\lbrack {k + \kappa_{1}} \right\rbrack}{b_{\nu}^{H}\left\lbrack {k + \kappa_{2}} \right\rbrack}} + {b_{\nu}^{H}\left\lceil {k + \kappa_{2}} \right\rceil b_{\nu}\left\lceil {k + \kappa_{1}} \right\rceil I}} \right)a\left\lceil {h + \kappa_{3}} \right\rceil h_{\nu}{\left\lceil \kappa \right\rceil.}}}}}}} & (47)\end{matrix}$

The time-domain and aliased frequency-domain kernel are related by athree-dimensional (3D) DTFT according to

h _(v)[k]=

⁻¹ {H _(v)(e^(jωT))}.   (48)

The kernel h_(v)[k]=h_(v)[k₁, k₂, k₃] is equivalent to the kernelderived via an integration over time and space in [10, Eq. (61), (62)]and used in [13].

Now, the relation to the GN-model and to system design rules isexplained.

Parseval's theorem applied to (48) yields

$\begin{matrix}{{E_{h,\nu}\overset{def}{=}{{\sum\limits_{\kappa \in {\mathbb{Z}}^{3}}{{h_{\nu}\lbrack\kappa\rbrack}}^{2}} = {\left( \frac{T}{2\pi} \right)^{3}{\int_{{\mathbb{T}}^{3}}{{{H_{\nu}\left( e^{j\omega T} \right)}}^{2}d^{3}\omega}}}}},} & (49)\end{matrix}$

where the right-hand side can be interpreted as an alternativeformulation of the (frequency-domain) Gaussian noise (GN)-model [27] in1/T-periodic continuous-frequency domain. In (49) the common pre-factor

$\left( {\frac{8}{9}\frac{L_{eff}}{L_{{NL},\nu}}} \right)^{2}$

is omitted here and the energy in time- and frequency domain iscalculated over the whole support of the probe and interferingwavelength channel, whereas [27, Eq. (1)] is evaluated only at a singlefrequency ω. Beyond that, to include all SCI and XCI contributions oneneeds to sum over all v—the GN-model in its standard form also includesMCI. This is the dual representation to the original work where theoptical signal is constructed as a continuous-time signal with period T₀and discrete frequency components (c.f. the Karhunen-Loève formula in[26], [34]). In other words, the discretization in one domain and theperiodicity in the other is exchanged in (49) compared to the GN-model.In this view, the result obtained by the GN-model corresponds to thekernel energy E_(h,v) of the corresponding end-to-end channel.

At the same time, the (system relevant) variance of the perturbation

$\sigma_{\Delta a}^{2}\overset{def}{=}{E\left\{ {{\Delta\; a}}^{2} \right\}}$

depends as well on the properties of the modulation format A which inturn is a problem addressed by the extended Gaussian noise (EGN)-model[34], cf. also the discussion in [5, Sec. F and Appx.]. Note, that thederivation of (49) does not require any assumptions on the signal(albeit its pulse-shape)—in particular no Gaussian assumption.

We can identify three relevant system parameters that characterize thenonlinear response: the map strength

_(T,ρ)=L_(eff)/L_(D) (or equivalently the v-dependent

_(T,v)=L_(eff)/L_(wo,v)) which is a measure of the temporal extent,i.e., the memory of the nonlinear interaction. Secondly, the(V-dependent) nonlinear phase shift

$\phi_{{NL},v}\overset{def}{=}{\frac{8}{9}\frac{L_{eff}}{L_{{\underset{\_}{N}L},\nu}}}$

that depends via L_(NL,v) linearly on the launch power P_(v) andessentially acts as a scaling factor to the nonlinear distortion Δα[k].And at last, the total kernel energy E_(h,v) which charactarizes thestrength of the nonlinear interaction—independent of the launch power.

Now, applications to fiber nonlinearity compensation according toembodiments is described.

The derived channel models also finds applications for fibernonlinearity compensation, where implementation complexity is ofparticular interest. An experimental demonstration of intra-channelfiber nonlinearity compensation based on the time-domain model in (38)has been presented in [35]. In terms of computational efficiency afrequency-domain implementation can be superior to the time-domainimplementation, in particular, for cases where the number of nonlinearinteracting pulses is large.

This is typically the case for large map strengths

_(T,≯), large relative frequency offsets Δω_(v), i.e., large

_(T,v), and pulse shapes h_(T)(t) that extend over multiple symboldurations, e.g., a root-raised cosine (RRC) shape with small roll-offfactor ρ. Then, the number of coefficients of the time-domain kernelh_(v)[k] exceeding a relevant energy level grows very rapidly leading toa large number of multiplications and summations. The frequency-domainpicture comprises only a double integral instead of a triple sum and canbe efficiently implemented using standard signal processing techniques.

Algorithm 1: REG-PERT-FD for the SCI contribution  1 a_(λ)[k] =overlapSaveSplit( 

a[k] 

, N_(DFT), K)  2 k, μ, μ₁, μ₂ ∈ {0, 1, . . . , N_(DFT) − 1}  3${H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack} = {{H_{\rho}\lbrack\mu\rbrack} = {H_{\rho}\left( e^{j\;\frac{2\;\pi}{N_{DFT}}\mu} \right)}}$ 4 forall λ do  5  A_(λ)[μ] = DFT{a_(λ)[k]}  6  forall μ do  7   μ₃ =mod_(N) _(DFT) (μ − μ₁ + μ₂)  8   ${\Delta\;{A_{\lambda}^{SCI}\lbrack\mu\rbrack}} = {{- j}\;\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}} \times {\sum_{\mu_{1},\mu_{2}}{{A_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{A_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$ 9   Y_(λ) ^(PERT)[μ] = A_(λ)[μ] + ΔA_(λ) ^(SCI)[μ] 10  end 11  y_(λ)^(PERT)[k] = DFT⁻¹{Y_(λ) ^(PERT)[μ]} 12 end 13

y^(PERT)[k] 

 = overlapSaveAppend(y_(λ) ^(PERT)[k], N_(DFT), K)

Exemplarily for the SCI contribution, Algorithm 1 realizes the regularperturbation (REG-PERT) procedure in 1/T-periodic discretefrequency-domain (FD) corresponding to the continuous-frequency relationin (38). Here, the overlap-save algorithm is used to split the sequence

α[k]

into overlapping blocks α_(λ)[k]

A_(λ)[μ] of size N_(DFT) enumerated by the subindex λ ∈

[36]. The block size is equal to the size of the discrete Fouriertransform (DFT) and the overlap between successive blocks is K. Theone-dimensional DFT is performed on each vector component of α_(λ)[k]and the correspondence always relates the whole blocks of length NDFT.

The aliased frequency-domain kernel is discretized to obtain thecoefficients

$\begin{matrix}{{H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack} = {{H_{\rho}\lbrack\mu\rbrack}\overset{def}{=}{H_{\rho}\left( e^{j\frac{2\pi}{N_{DFT}}\mu} \right)}}} & (50)\end{matrix}$

where NDFT is the number of discrete-frequency samples. Thediscrete-frequency indices μ₁ and μ₂ are elements of the set {0, 1, . .. , N_(DFT)−1} whereas μ₃ must be (modulo) reduced to the same numberset due to the 1/T-periodicity of ω₃ in (37). The number of coefficientscan be decreased by pruning, similar to techniques already applied toVSTF models [37]. However, note that in contrast to VSTF models theproposed algorithm operates on the 1/T-periodic spectrum of blocks oftransmit symbols a_(λ)[k] and the filter coefficients are taken from thealiased frequency-domain kernel. Line 8 of the algorithm effectivelyrealizes equation (37) where the (double) sum is performed over all μ₁and μ₂* After frequency-domain processing the blocks of perturbedreceive symbols Y_(λ) ^(PERT)[μ]

y_(λ) ^(PERT)[k] are transformed back to time domain where the N_(DFT)-Kdesired output symbols of each block are appended to obtain theperturbed sequence

y^(PERT)[k]

. Algorithm 1 can be generalized to XCI analogously to (46).

According to an embodiment, Algorithm 1 for XCI reads as follows:

Algorithm 1: REG-PERT-FD for the XCI contribution of the v^(th)wavelength channel  1 a_(λ)[k] = overlapSaveSplit( 

a[k] 

, N_(DFT), K)  2 b_(λ)[k] = overlapSaveSplit( 

b_(v)[k] 

, N_(DFT), K)  3 k, μ, μ₁, μ₂ ∈ {0, 1, . . . , N_(DFT) − 1}  4${H_{v}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack} = {{H_{v}\lbrack\mu\rbrack} = {H_{v}\left( e^{j\;\frac{2\;\pi}{N_{DFT}}\mu} \right)}}$ 5 forall λ do  6  A_(λ)[μ] = DFT{a_(λ)[k]}  7  B_(λ)[μ] = DFT{b_(λ)[k]} 8  forall μ do  9   μ₃ = mod_(N) _(DFT) (μ − μ₁ + μ₂) 10   ${\Delta\;{A_{\lambda}^{XCI}\lbrack\mu\rbrack}} = {{- j}\;\frac{\phi_{{NL},v}}{N_{DFT}^{2}} \times {\sum_{\mu_{1},\mu_{2}}{\left( {{{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}} + {{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}I}} \right) \times {A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{v}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$11   Y_(λ) ^(PERT)[μ] = A_(λ)[μ] + ΔA_(λ) ^(XCI)[μ] 12  end 13  y_(λ)^(PERT)[k] = DFT⁻¹{Y_(λ) ^(PERT)[μ]} 14 end 15

y^(PERT)[k] 

 = overlapSaveAppend(y_(λ) ^(PERT)[k], N_(DFT), K)

The time- and frequency-domain picture of the regular perturbationapproach are equivalent due to the DTFT in (37), (38) which interrelatesboth representations. Algorithm 1 represents a practical realization indiscrete-frequency which produces the same (numerical) results as thediscrete-time model as long as N_(DFT) and K are chosen sufficientlylarge for a given system scenario. To that end, below, the regulardiscrete-time and -frequency model and the reference channel modelimplemented via the SSFM are compared. Then, the regular model isextended to a combined regular-logarithmic model where a subset of theperturbations are considered as multiplicative, i.e., perturbations thatcause a rotation in phase or in the state of polarization (SOP).

Now, a regular-logarithmic model in the discrete-time domain isprovided.

It was already noted in [38] that the regular VSTF approach (or theequivalent RP method) in (26) reveals an energy-divergence problem ifthe optical launch power P is too high—or more precisely if thenonlinear phase shift ϕ_(NL) is too large.. Using a first-order RPapproach, a pure phase rotation is approximated by exp(jϕ)≈1+jϕ. Whilemultiplication with exp(jϕ) is an energy conserving transformation(i.e., the norm is invariant under phase rotation), the RP approximationis obviously not energy conserving. In the context of opticaltransmission, already a trivial (time-constant) average phase rotationdue nonlinear interaction is not well modeled by the RP method.

This inconsistency was first addressed in the early 2000s [4], [39] andyears later revived in the context of intra-channel fiber nonlinearitymitigation. E.g. in [40], [41] it turned out that a certain subset ofsymbol combinations in the time-domain RP model deterministicallycreates a perturbation oriented into the -j-direction from the transmitsymbol a[k]. Similarly, in the pulse-collision picture [11]-[13] asubset of degenerate cross-channel pulse collisions were properlyassociated to distortions exhibiting a multiplicative nature. In thesame series of contributions, these subsets of degenerate, in the sensethat not all four interacting pulses are distinct, distortions werefirst termed two- and three-pulse collisions, i.e., symbol combinationsk ∈

³ in (47) with k₃=0 in our terminology. While the pulse collisionpicture covers only cross-channel effects, we will extent the analysisalso to intra-channel effects.

In this context, we review some properties of the kernel coefficientsrelevant for inter-channel (v≠ρ) two- and three-pulse collisions [13]

h_(v)[k₁, k₂, 0]∈

. if k₁=k₂   (51)

h_(v)[k₁, k₂, 0]=h_(v)*[k₂, k₁, 0]∈

if k₁≠k₂,   (52)

where two-pulse collisions with k₁=k₂ in (51) are doubly degenerate andthe kernel is real-valued. The transmit pulse-shape h_(T) (t) is assumedto be a real-valued (root) raised-cosine.

In case of three-pulse collisions, the kernel is generallycomplex-valued but due to its symmetry property in (52) and the doublesum over all (nonzero) pairs of [k₁, k₂]^(T) in (47) the overall effectis still multiplicative.

Additionally, for intra-channel contributions (v=ρ) we find thefollowing symmetry properties of the kernel

h_(ρ)[k₁, k₂, k₃]=h_(ρ)[k₃, k₂, k₁]  (53)

h_(ρ)[k₁, k₂, k₃]=h_(ρ)[-k₁, -k₂, -k₃],   (54)

and we identify a second degenerate case with k₁=0 as source formultiplicative distortions, cf. the symmetric form of (38) w.r.t. k₁ andk₃.

In the following, the original RP solution is modified such thatperturbations originating from certain degenerate mixing products areassociated with a multiplicative perturbation. Similar to [13], [41],[42], we extend the previous RP model to a combined regular-logarithmicmodel. It takes the general form of

y[k]=exp (jΦ[k]+j{right arrow over (s)}[k]·{right arrow over (σ)})(a[k]+Δa[k]).   (55)

In addition to the regular, additive perturbation Δa[k] we now alsoconsider a phase rotation by exp(jΦ[k]) and a rotation in the state ofpolarization by exp(j{right arrow over (s)}[k]·{right arrow over (σ)}).Here, exp(·) denotes the matrix exponential. All perturbative termscombine both SCI and XCI effects, i.e., the additive perturbation Δa[k]∈

² is the sum of SCI and XCI contributions. The time-dependent phaserotation is given by exp(jΦ[k]) with the diagonal matrix Φ[k] ∈

^(2×2) defined as

$\begin{matrix}{{{\Phi\lbrack k\rbrack}\overset{def}{=}{{{\phi^{SCI}\lbrack k\rbrack}I} + {{\phi^{XCI}\lbrack k\rbrack}I}}},} & (56)\end{matrix}$

i.e., we find a common phase term for both polarizations originatingfrom intra- and inter-channel effects. The combined effect of intra- andinter-channel cross-polarization modulation (XPoIM) is expressed by thePauli matrix expansion {right arrow over (s)}[k]·{right arrow over (σ)}∈

^(2×2) using (6), with the notation adopted from [20] and [43]. Theexpansion defines a unitary rotation in Jones space of the perturbedvector a[k]+Δa[k] around the time-dependent Stokes vector {right arrowover (s)}[k] and is explained in more detail in the following.

1) SCI Contribution: TAT o discuss the SCI contribution we firstintroduce the following symbol sets

S ⁢ C ⁢ I ⁢ = def ⁢ { [ κ 1 , κ 2 , κ 3 ] T ∈ ℤ 3 ⁢   h ρ ⁡ [ κ ] / h ρ ⁡ [ 0]  2 > Γ SCI } ( 57 ) ϕ ⊕ ⁢ = def ⁢ S ⁢ C ⁢ I  ⁢ κ 1 = 0 ⩓ κ 2 ≠ 0 ⩓ κ 3 ≠0 } ( 58 ) ϕ ⊖ ⁢ = def ⁢ { S ⁢ C ⁢ I ⁢  ⁢ κ 3 = 0 ⩓ κ 2 ⁢ ≠ 0 ⩓ κ 1 ≠ 0 } ( 59) ϕ S ⁢ C ⁢ I ⁢ = def ⁢ ϕ ⊕ ⋂ ϕ ⊖ ⋂ { κ = 0 } ( 60 ) Δ S ⁢ C ⁢ I ⁢ = def ⁢ S ⁢ C ⁢I ⁢ ∖ ϕ S ⁢ C ⁢ I , ( 61 )

where (57) defines the base set including all possible symbolcombinations that exceed a certain energy level Γ^(SCI) normalized tothe energy of the center tap at k=0. In (58), (59) the joint set ofdegenerate two- and three-pulse collisions for SCI are defined whichfollow directly from the kernel properties in (51),(52) for k₃=0, and(53),(54) for k₁=0. The set of indices for multiplicative distortionsK_(ϕ) ^(SCI) in (60) also includes the singular case k=0. Then, theadditive set is simply the complementary set of K_(ϕ) ^(SCI)) w.r.t. thebase set

^(SCI).

We start with the additive perturbation defined above in (38) which nowreads

Δ ⁢ ⁢ a SCI ⁡ [ k ] = - j ⁢ ϕ NL , ρ ⁢ ∑ Δ S ⁢ C ⁢ I ⁢ a ⁡ [ k + κ 1 ] ⁢ a H ⁡ [k + κ 2 ] ⁢ a ⁡ [ k + κ 3 ] ⁢ h ρ ⁡ [ κ ] , ( 62 )

where the triple sum is now restricted to the set K_(Δ) ^(SCI) excludingall combinations which result in a multiplicative distortion, cf. (61).

To calculate the common phase ϕ^(SCI)[k] and the intra-channel Stokesrotation vector {right arrow over (s)}^(SCI)[k] we first analyse theexpression a[k+k₁]a^(H)[k+k₂]a[k+k₃] from the original equation in (38).For the set K_(ϕ) ^(⊕) with k₁=0 the triple product factors into therespective transmit symbol a[k] and a scalar value a^(H)[k+k₂]a[k+k₃].After multiplication with h_(ρ)[0, k₂, k₃] and summation of all k ∈K_(ϕ)^(⊕) the perturbation is strictly imaginary-valued (cf. symmetryproperties in (53),(54)).

On the other hand, for K_(ϕ) ^(↑) with k₃=0 we have to rearrange thetriple product using the matrix expansion from (7) to factor theexpression accordingly as16

aa^(H)a=½(a^(H)a I+(a^(H){right arrow over (σ)}a)·{right arrow over(σ)}) a.   (63)

(multiplication with h_(ρ)[k] and summation over k ∈

_(ϕ) ^(SCI) are implied)

The first term a^(H)a I also contributes to a common phase term, whereasthe second term (a^(H){right arrow over (σ)}a)·{right arrow over (σ)} ∈

^(2×2) is a traceless and Hermitian matrix exp(j(a^(H){right arrow over(σ)}a)·{right arrow over (σ)}) is a unitary polarization rotation. Sincethe Pauli expansion {right arrow over (μ)}·{right arrow over (σ)} in (6)is Hermitian, the expression exp(j {right arrow over (u)}·{right arrowover (σ)}) is unitary.

The multiplicative perturbation exp(jϕ^(SCI)[k] with ϕ^(SCI)[k] ∈

is then given by

ϕ S ⁢ C ⁢ I ⁡ [ k ] = ⁢ - ϕ NL , ρ ⁢ ∑ ϕ ⊕ ⁢ a H ⁡ [ k + κ 2 ] ⁢ a ⁡ [ k + κ 3 ] ⁢h ρ ⁡ [ κ ] - ⁢ 1 2 ⁢ ϕ NL , ρ ⁢ ∑ ϕ ⊖ ⁢ a H ⁡ [ k + κ 2 ] ⁢ a ⁡ [ k + κ 1 ] ⁢ hρ ⁡ [ κ ⁢ ] - ⁢ ϕ NL , ρ ⁢  a ⁡ [ k ]  2 ⁢ h ρ ⁡ [ 0 ] ( 64 ) = ⁢ - 3 2 ⁢ ϕ NL, ρ ⁢ ∑ ϕ ⊖ ⁢ a H ⁡ [ k + κ 2 ] ⁢ a ⁡ [ k + κ 1 ] ⁢ h ρ ⁡ [ κ ] - ⁢ ϕ NL , ρ ⁢ a ⁡ [ k ]  2 ⁢ h ρ ⁡ [ 0 ] . ( 65 )

Given a wide-sense stationary transmit sequence

a[k]

, the induced nonlinear phase shift has a time-average value ϕ^(SCI),around which the instantaneous phase ϕ^(SCI)[k] may fluctuate (cf. also[44]).

The instantaneous rotation of the SOP due to the expression exp(j{rightarrow over (s)}^(SCI)[k]·{right arrow over (σ)}) ∈C^(2×2) causesintra-channel XPoIM [45]. It is given by

s → S ⁢ C ⁢ I ⁡ [ k ] · σ → = ⁢ - 1 2 ⁢ ( ϕ NL , ρ ) ⁢ ∑ ϕ ⊖ ⁢ ( 2 ⁢ a ⁡ [ k + κ1 ] ⁢ a H ⁡ [ k + κ 2 ] - ⁢ a H ⁡ [ k + κ 2 ] ⁢ a ⁡ [ k + κ 1 ] ⁢ I ) ⁢ h ρ ⁡ [ κ] , ( 66 )

where we made use of the relation in (6). The rotation matrixexp(j{right arrow over (s)}^(SCI)[k]·{right arrow over (σ)}) is unitaryand {right arrow over (s)}^(SCI)[k]·{right arrow over (σ)} is Hermitianand traceless. The physical meaning of the transformation described in(66) is as follows: The perturbed transmit vector (a[k]+Δa[k]) in (55)is transformed into the polarization eigenstate {right arrow over(s)}[k] (i.e., into the basis defined by the eigenvectors of {rightarrow over (s)}^(SCI)[k]·{right arrow over (σ)}). There, both vectorcomponents receive equal but opposite phase shifts and the result istransformed back to the x/y-basis of the transmit vector. In Stokesspace, the operation can be understood as a precession of ({right arrowover (a)}[k]+{right arrow over (a)}[k]) around the Stokes vector {rightarrow over (s)}^(SCI)[k] by an angle equal to its length ∥{right arrowover (s)}^(SCI)[k]. The intra-channel Stokes vector {right arrow over(s)}^(SCI)[k] depends via the nonlinear kernel h_(ρ)[k] on the transmitsymbols within the memory of the nonlinear interaction S_(T,ρ) arounda[k]. Similar to the nonlinear phase shift—for a wide-sense stationaryinput sequence—the Stokes vector {right arrow over (s)}^(SCI)[k] has atime-constant average value around which it fluctuates over time.

2) XCI Contribution: The same methodology is now applied tocross-channel effects. The symbol set definitions for XCI follow fromthe considerations described above.

ν X ⁢ C ⁢ i = { [ κ 1 , κ 2 , ⁢ κ 3 ] T ∈ ℤ 3 ⁢   h ν ⁡ [ κ ] / h ν ⁡ [ 0 ] 2 > Γ ν XCI } ⁢ ⁢ ϕ , ν X ⁢ C ⁢ I ⁢ = def ⁢ d · f ⁢ { K ν XCI | κ 3 = 0 ⩓ κ 2≠ 0 ⩓ κ 1 ≠ 0 } ( 67 ) ⋂ { κ = 0 } ( 68 ) Δ , ν X ⁢ C ⁢ I ⁢ = def ⁢ ν XCI ∖ϕ , ν X ⁢ C ⁢ I , ( 69 )

where the subscript v indicates the channel number of the respectiveinterfering channel. For

_(ϕ,v) ^(XCI), only the degenerate case

₃=0 has to be considered due to the kernel properties of h_(v)[k₁, k₂,0] in (51),(52). Similar to (63), the expression bb^(H)+b^(H)b I from(47) is rearranged to obtain

$\begin{matrix}{{\frac{3}{2} - \underset{b^{H}{bI}}{\underset{︸}{\begin{bmatrix}{{b_{x}b_{x}^{*}} + {b_{y}b_{y}^{*}}} & 0 \\0 & {{b_{y}b_{y}^{*}} + {b_{x}b_{x}^{*}}}\end{bmatrix}}} + {\frac{1}{2}\underset{{{2{bb}^{H}} - {b^{H}{bI}}} = {{({b^{h}\overset{\rightarrow}{\sigma}b}\;)} \cdot \overset{\rightarrow}{\sigma}}}{\underset{︸}{\begin{bmatrix}{{b_{x}b_{x}^{*}} + {b_{y}b_{y}^{*}}} & {2b_{x}b_{y}^{*}} \\{2b_{y}b_{x}^{*}} & {{b_{y}b_{y}^{*}} + {b_{x}b_{x}^{*}}}\end{bmatrix}}}}},} & (70)\end{matrix}$

where the argument and subscript v is omitted for concise notation. Themultiplicative cross-channel contribution is again split into a commonphase shift in both polarizations and an equal but opposite phase shiftin the basis given by the instantaneous Stokes vector of the v^(th)interferer. We define the total, common phase shift due to cross-channelinterference as

ϕ XCI ⁡ [ k ] = - ∑ ν ≠ ρ ⁢ 3 2 ⁢ ϕ N ⁢ L , ν ⁢ ∑ ϕ , ν XCI ⁢ b ν H ⁡ [ k + κ 1] ⁢ b ν ⁡ [ k + κ 2 ] ⁢ h ν ⁡ [ κ ] ( 71 )

which depends on the instantaneous sum over all interfering channels andthe sum of b_(v) ^(H)b_(v) bv over [k₁, k₂]^(T). The effective,instantaneous cross-channel Stokes vector {right arrow over (s)}[k] isgiven by

s → XCI ⁡ [ k ] · σ → = ⁢ - ∑ ν ≠ p ⁢ 1 2 ⁢ ϕ N ⁢ L , ν ⁢ ∑ ϕ , ν XCI ⁢ ( 2 ⁢ bν ⁡ [ k + κ 1 ] ⁢ b ν H ⁡ [ k + κ 2 ] - ⁢ b ν H ⁡ [ k + κ 2 ] ⁢ b ν ⁡ [ k + κ 1] ⁢ I ) ⁢ h ν ⁡ [ κ ] . ( 72 )

Note, that the expressions in (71), (72) include both contributions fromtwo- and three pulse collisions (cf. [13, Eq. (10)- (13)]).

3) Energy of Coefficients in Discrete-Time Domain: The energy of thekernel coefficients is defined according to Parseval's theorem in (49)for the subsets given in (57-61). We find for the different symbol sets

E h SCI ⁢ = def ⁢ ∑ ϕ S ⁢ C ⁢ I ⁢  h ρ ⁡ [ κ ]  2 ( 73 ) E h , Δ S ⁢ C ⁢ I ⁢ =def ⁢ ∑ ϕ S ⁢ C ⁢ I ⁢  h ρ ⁡ [ κ ]  2 ( 74 ) E h , ϕ S ⁢ C ⁢ I ⁢ = def ⁢ ∑ ϕ S ⁢C ⁢ I ⁢  h ρ ⁡ [ κ ]  2 , ( 75 )

with the clipping factor Γ^(SCI) in (57) equal to zero. The energy forcross-channel effects is defined accordingly with the sets from (67-69).Since the subsets for additive and multiplicative effects are alwaysdisjoint we have E_(h)=E_(h,Δ)+E_(h,ϕ).

Now, a regular-logarithmic model in frequency domain is provided.

Similar to the above, we first review some kernel properties of thealiased frequency-domain kernel coefficients

$\begin{matrix}\begin{matrix}{{{H_{v}\left( e^{j\omega T} \right)} \in {\mathbb{R}}},} & {{{{if}\mspace{14mu}\omega_{2}} = {\left. \omega_{1}\Leftrightarrow\omega_{3} \right. = {\left. \omega\Leftrightarrow v_{2} \right. = 0}}},}\end{matrix} & (76) \\\begin{matrix}{{{H_{\rho}\left( e^{j\;\omega\; T} \right)} \in {\mathbb{R}}},} & {{{if}\mspace{14mu}\omega_{2}} = {\left. \omega_{1}\Leftrightarrow\omega_{3} \right. = {\left. \omega\Leftrightarrow v_{2} \right. = 0}}} \\\; & {{{\bigvee\omega_{2}} = {\left. \omega_{3}\Leftrightarrow\omega_{1} \right. = {\left. \omega\Leftrightarrow v_{1} \right. = 0}}},}\end{matrix} & (77)\end{matrix}$

where the two (doubly) degenerate cases ω₁=ω₂ and ω₃=ω₂ correspond toclassical inter- and intra-channel cross-phase modulation (XPM).Accordingly, the frequency domain model is now modified such that thesecontribution will be associated with multiplicative distortions.However, due to the multiplicative nature, only average values can beincorporated into the frequency-domain model as they are both constantover time and frequency and can be treated as a common pre-factor inboth pictures. We will see in the following that this already leads tosignificantly improved results compared to the regular model. Note that,in contrast to the regular models, the regular-logarithmic model in timeand frequency are no longer equivalent.

The general form of the combined regular-logarithmic model in frequencyis given by

Y(e ^(jωT))=exp (j{right arrow over (Φ)}+J{right arrow over (S)}·{rightarrow over (σ)})×(A(e ^(jωT))+ΔA(e ^(jωT))),   (78)

where the phase- and polarization-term take on a frequency-constantvalue, i.e., independent of e^(jωT) (and vice-versa independent of k inthe time-domain picture). Following the same terminology as before, weintroduce the average multiplicative perturbation of the common phaseterm

$\begin{matrix}{{\overset{\_}{\Phi}\overset{def}{=}{{{\overset{\_}{\phi}}^{SCI}I} + {{\overset{\_}{\phi}}^{XCI}I}}},} & (79)\end{matrix}$

as the sum of the intra-channel contribution ϕ ^(SCI) ∈

and the inter-channel contribution ϕ ^(XCI) ∈

. Similarly, for the average polarization rotation we have

$\begin{matrix}{{{\overset{\rightarrow}{S} \cdot \overset{\rightarrow}{\sigma}}\overset{def}{=}{{{\overset{\rightarrow}{S}}^{SCI} \cdot \overset{\rightarrow}{\sigma}} + {{\overset{\rightarrow}{S}}^{XCI} \cdot \overset{\rightarrow}{\sigma}}}},} & (80)\end{matrix}$

where {right arrow over (S)}·{right arrow over (σ)} is again Hermitianand traceless, which in turn makes the matrix exponential exp(j{rightarrow over (S)}·{right arrow over (σ)}) unitary.

1) SCI Contribution: The two degenerate frequency conditions in (77) areused in the expression (37) to obtain the average, intra-channel phasedistortion. To that end, the triple product AA^(H)A in (37) isrearranged similar to (63). First, the general frequency-dependentexpression ϕ^(SCI)(e^(jωT)) is given by

$\begin{matrix}\begin{matrix}{\phi^{SCI} = {{- \phi_{{NL},\rho}}\frac{T}{\left( {2\pi} \right)^{2}}{\int_{\mathbb{T}}{{{A\left( e^{j\omega_{2}T} \right)}}^{2}{H_{p}\left( e^{{j{\lbrack{\omega,\omega_{2},\omega_{2}}\rbrack}}^{T}T} \right)}d\;\omega_{2}}}}} \\{{= {{- \frac{1}{2}}\phi_{{NL},\rho}\frac{T}{\left( {2\pi} \right)^{2}}{\int_{\mathbb{T}}{{{A\left( d^{j\;\omega_{1}T} \right)}}^{2}{H_{p}\left( e^{{j{\lbrack{\omega_{1},\omega_{1},\omega}\rbrack}}^{T}T} \right)}d\omega_{1}}}}},}\end{matrix} & \left( {81a} \right)\end{matrix}$

where the first term on the right-hand side in (81) corresponds to thedegeneracy ω₂=ω₃⇔ω₁=ω and the second term corresponds to ω₂=ω₁⇔ω₃=ω. Wesimplify the expression using the RRC σ=0 approximation to obtain theaverage, intra-channel phase distortion

$\begin{matrix}{{{\overset{\_}{\phi}}^{SCI} = {\frac{3}{2}\phi_{{NL},\rho}\frac{T}{\left( {2\pi} \right)^{2}}{\int_{\mathbb{T}}{{{A\left( e^{j\;\omega\; T} \right)}}^{2}d\omega}}}},} & \left( {81b} \right)\end{matrix}$

which does no longer depend on the power or dispersion profile of thetransmission link (given a fixed Leff).

Similarly, the average intra-channel XPoIM contribution can besimplified to

$\begin{matrix}\begin{matrix}{{{\overset{\rightarrow}{S}}^{SCI} \cdot \overset{\rightarrow}{\sigma}} = {\frac{1}{2}\phi_{{NL},\rho}\frac{T}{\left( {2\pi} \right)^{2}}{\int_{\mathbb{T}}\left( {{2{A\left( e^{j\;\omega\; T} \right)}{A^{H}\left( e^{j\;\omega\; T} \right)}} -} \right.}}} \\{\left. {{A^{H}\left( e^{j\;\omega\; T} \right)}{A\left( e^{j\;\omega\; T} \right)}I} \right)d\;{\omega.}}\end{matrix} & (88)\end{matrix}$

In Algorithm 2 the used modifications to the regular perturbation model(REG-PERT) are highlighted to arrive at the regular-logarithmicperturbation model (REGLOG-PERT)-again exemplarily for the SCIcontribution. Lines 6,7 of Algorithm 2 translate Eq. (81a), (81b), (82)to the discrete-frequency domain where the integral over all ω ∈

becomes a sum over all μ of the λ^(th) processing block. The averagevalues, here, are always associated to the average values of the λ^(th)block. In Lines 10,11, the double sum to obtain ΔA_(λ) ^(SCI)[μ] isrestricted to all combinations U of the discrete frequency pair [μ₁,μ₂]^(T) excluding the degenerate cases corresponding to Eq. (76), (77).The perturbed receive vector Y_(λ) ^(PERT) is then calculated accordingto (78) before it is transformed back to the discrete-time domain.

2) XCI Contribution: The cross-channel contributions follow from theconsiderations above and we obtain for the degenerate case in (76) thetotal, average XCI phase shift

$\begin{matrix}{{\overset{¯}{\phi}}^{XCI} = {- {\sum\limits_{\nu \neq \rho}\;{\frac{3}{2}\phi_{{NL},\nu}\frac{T}{\left( {2\pi} \right)^{2}}{\int_{\mathbb{T}}{{{B_{\nu}\left( d^{j\;\omega\; T} \right)}}^{2}d\;\omega}}}}}} & (83)\end{matrix}$

and analogously for the total, average XCI Stokes vector we find

$\begin{matrix}\begin{matrix}{{{\overset{\rightarrow}{S}}^{XCI} \cdot \overset{\rightarrow}{\sigma}} = {- {\sum\limits_{\nu \neq \rho}{\frac{1}{2}\phi_{{NL},{\nu\frac{T}{{({2\pi})}^{2}}}}{\int_{\mathbb{T}}\left( {{2{B_{\nu}\left( e^{j\omega T} \right)}{B_{\nu}^{H}\left( e^{j\omega T} \right)}} -} \right.}}}}} \\{\left. {{B_{\nu}^{H}\left( e^{j\omega T} \right)}{B_{\nu}\left( e^{j\omega T} \right)}I} \right)d\;{\omega.}}\end{matrix} & (84)\end{matrix}$

3) Energy of Coefficients in Discrete-Frequency Domain: With thenotation of the discrete-frequency kernel

${H_{\nu}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack} = {{H_{v}\lbrack\mu\rbrack} = {H_{\nu}\left( e^{j\frac{2\pi}{N_{DFT}}\mu} \right)}}$

we have the following definitions

$\begin{matrix}{E_{H}^{SCI}\overset{def}{=}{\frac{1}{N_{DFT}^{3}}{\sum\limits_{\mathcal{U}^{SCI}}{{H_{\rho}\lbrack\mu\rbrack}}^{2}}}} & (85) \\{E_{H,\Delta}^{SCI}\overset{def}{=}{\frac{1}{N_{DFT}^{3}}{\sum\limits_{\mathcal{U}_{\Delta}^{SCI}}{{H_{\rho}\lbrack\mu\rbrack}}^{2}}}} & (86) \\{{E_{H,\phi}^{SCI}\overset{def}{=}{\frac{1}{N_{DFT}^{3}}{\sum\limits_{\mathcal{U}_{\phi}^{SCI}}{{H_{\rho}\lbrack\mu\rbrack}}^{2}}}},} & (87)\end{matrix}$

Following the regular-logarithmic approach, some of the degeneratedistortion should be associated to multiplicative distortions. In thecontext of fiber nonlinearity compensation, these terms correspond to anonlinear-induced phase distortion or a nonlinear-induced distortion ofthe state of polarization. These distortions can be compensated for byapplying the inverse operation on the transmit or receive-side, e.g.,mathematically speaking by changing the sign in the exponential in (55).The (frequency-domain) intra-channel phase distortion term can becalculated according to (81a) and (81b) while the polarizationdistortion term is calculated according to (82). The inter-channel termsare given in (83) and (84).

In the following, Algorithm 2 (REGLOG-PERT-FD) f_(o)r the SCIcontribution is provided:

Algorithm 2: REGLOG-PERT-FD for the SCI contribution  1 a_(λ)[k] =overlapSaveSplit( 

a[k] 

, N_(DFT), K)  2 k, μ, μ₁, μ₂ ∈ {0, 1, . . . , N_(DFT) − 1}  3${H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack} = {{H_{\rho}\lbrack\mu\rbrack} = {H_{\rho}\left( e^{j\;\frac{2\pi}{N_{DFT}}\mu} \right)}}$ 4 forall λ do  5  A_(λ)[μ] = DFT{a_(λ)[k]}  6  ${\overset{\_}{\phi}}_{\lambda}^{SCI} = {{- \frac{3}{2}}\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}}{\sum_{\mu}{{A_{\lambda}\lbrack\mu\rbrack}}^{2}}}$ 7  ${{\overset{\rightarrow}{S}}_{\lambda}^{SCI} \cdot \overset{\rightarrow}{\sigma}} = {{{- \frac{1}{2}}\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}}{\sum_{\mu}{2\;{A_{\lambda}\lbrack\mu\rbrack}{A_{\lambda}^{H}\lbrack\mu\rbrack}}}} - {{{A_{\lambda}\lbrack\mu\rbrack}}^{2}I}}$ 8  forall μ do  9   μ₃ = mod_(N) _(DFT) (μ − μ₁ + μ₂) 10   U = {[μ₁,μ₂]^(T) | μ₂ ≠ μ₁ ∧ μ₂ ≠ μ₃} 11   ${\Delta\;{A_{\lambda}^{SCI}\lbrack\mu\rbrack}} = {{- j}\;\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}} \times {\sum_{u}{{A_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{A_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$12   Y_(λ) ^(PERT)[μ] = exp(jϕ _(λ) ^(SCI)I + j{right arrow over(S)}_(λ) ^(SCI) · {right arrow over (σ)}) ×   (A_(λ)[μ] + ΔA_(λ)^(SCI)[μ]) 13  end 14  y_(λ) ^(PERT)[k] = DFT⁻¹{Y_(λ) ^(PERT)[μ]} 15 end16

y^(PERT)[k] 

 = overlapSaveAppend(y_(λ) ^(PERT)[k], N_(DFT), K)

with the sets according to (77)

^(SCI)={μ=[μ₁, μ₂,μ₃]^(T) ∈{0,1, . . . , N_(DFT)-1}³}   (88)

U_(Δ) ^(SCI)={U^(SCI)|μ₂≠μ₁∧μ₂≠μ₃}   (89)

U_(ϕ) ^(SCI)={U^(SCI)|μ₂=μ₁∨μ₂=μ₃}.   (90)

Note, that we have again E_(H) ^(SCI)=E_(H,Δ) ^(SCI)+E_(H,ϕ) ^(SCI) anddue to Parseval's theorem E_(h) ^(SCI)=E_(H) ^(SCI) for N_(DFT)→∞. Thecardinalities of the sets are

^(SCI)|=N_(DFT) ³. |U_(ϕ) ^(SCI)=2N_(DFT) ² and |U_(Δ)^(SCI)|=|I^(SCI)|−|U_(ϕ) ^(SCI)|. The cross-channel sets are definedaccording to (76) with only a single degeneracy.

In an embodiment, algorithm 2 for XCI reads as follows:

Algorithm 2: REGLOG-PERT-FD for the XCI contribution of the v^(th)wavelength channel  1 a_(λ)[k] = overlapSaveSplit( 

a[k] 

, N_(DFT), K)  2 b_(λ)[k] = overlapSaveSplit( 

b_(v)[k] 

, N_(DFT), K)  3 k, μ, μ₁, μ₂ ∈ {0, 1, . . . , N_(DFT) − 1}  4${H_{v}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack} = {{H_{v}\lbrack\mu\rbrack} = {H_{v}\left( e^{j\;\frac{2\pi}{N_{DFT}}\mu} \right)}}$ 5 forall λ do  6  A_(λ)[μ] = DFT{a_(λ)[k]}  7  B_(λ)[μ] = DFT{b_(λ)[k]} 8  ${\overset{\_}{\phi}}_{\lambda}^{XCI} = {{- \frac{3}{2}}\frac{\phi_{{NL},v}}{N_{DFT}^{2}}{\sum_{\mu}{{B_{\lambda}\lbrack\mu\rbrack}}^{2}}}$ 9  ${{\overset{\rightarrow}{S}}_{\lambda}^{XCI} \cdot \overset{\rightarrow}{\sigma}} = {{{- \frac{1}{2}}\frac{\phi_{{NL},v}}{N_{DFT}^{2}}{\sum_{\mu}{2\;{B_{\lambda}\lbrack\mu\rbrack}{B_{\lambda}^{H}\lbrack\mu\rbrack}}}} - {{{B_{\lambda}\lbrack\mu\rbrack}}^{2}I}}$10  forall μ do 11   μ₃ = mod_(N) _(DFT) (μ − μ₁ + μ₂) 12   U = {[μ₁,μ₂]^(T) | μ₂ ≠ μ₁} 13   ${\Delta\;{A_{\lambda}^{XCI}\lbrack\mu\rbrack}} = {{- j}\;\frac{\phi_{{NL},v}}{N_{DFT}^{2}} \times {\sum_{u}{\left( {{{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}} + {{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}I}} \right) \times {A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{v}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$14   Y_(λ) ^(PERT)[μ] = exp(jϕ _(λ) ^(XCI)I + j{right arrow over(S)}_(λ) ^(XCI) · {right arrow over (σ)}) ×   (A_(λ)[μ] + ΔA_(λ)^(XCI)[μ]) 15  end 16  y_(λ) ^(PERT)[k] = DFT⁻¹{Y_(λ) ^(PERT)[μ]} 17 end18

y^(PERT)[k] 

 = overlapSaveAppend(y_(λ) ^(PERT)[k], N_(DFT), K)

In the following, numerical results are provided.

The following complements the general considerations of the above bynumerical simulations. To this end, we compare the simulated receivedsymbol sequence <y[k]> obtained by the perturbation-based (PERT)end-to-end channel models to the sequence obtained by numericalevaluation via the SSFM (in the following indicated by the superscriptSSFM).

The evaluated metric is the normalized MSE between the two outputsequences for a given input sequence

a[k]

, i.e., we have

$\begin{matrix}{{\sigma_{e}^{2}\overset{def}{=}{E\left\{ {{y^{SSFM} - y^{PERT}}}^{2} \right\}}},} & (91)\end{matrix}$

where the expectation takes the form of a statistical average over thetime of the received sequence. The MSE is already normalized due to thefixed variance σ_(a) ²=1 of the symbol alphabet and the receiver-sidere-normalization in (25), s.t. the received sequence has (approximately19) the same fixed variance as the transmit sequence.

19In the numerical simulation via SSFM signal depletion takes place dueto an energy transfer from signal to NLI. For simplicity, thisadditional signal energy loss is not accounted for by additionalreceiver-side re-normalization.

The simulation parameters are summarized in Table I. A total number ofN_(SYM)=¹⁶ transmit symbols

a[k]

are randomly drawn from a polarization-division multiplex (PDM) 64-aryquadrature amplitude modulation (QAM) symbol alphabet

with (4D) cardinality M=|

=4096, i.e., 64-QAM per polarization. The transmit pulse shape h_(T)(t)is a RRC with roll-off factor ρ and energy E_(T), to vary the opticallaunch power P. Above, signals have been treated as dimensionlessentities, but by convention we will still associate the optical launchpower P with units of [W] and the nonlinearity coefficient γ with[1/(Wm)].

Two different optical amplification schemes are considered: idealdistributed Raman amplification (i.e., lossless transmission) andtransparent end-of-span lumped amplification (i.e., lumped amplificationwhere the effect of signal-gain depletion [5, Sec. II B.] is neglectedin the derivation of the perturbation model). For lumped amplificationwe consider homogeneous spans of SSMF with fiber attenuation 10log₁₀=0.2 dB/km and a span length of L_(sp)=100 km In case of losslesstransmission we have 10 log₁₀e^(α)=0 dB/km and span length L_(sp)=21.71km corresponding to the asymptotic effective length

$L_{{eff},a}\overset{def}{=}{1/\alpha}$

or a TIUMUUS IIIJUF with infinite length and attenuation 10 log₁₀e^(α)=0.2 dB/km. The dispersion profile

_((z))=β₂z conforms with modern dispersion uncompensated (DU) links,i.e., without optical inline dispersion compensation and bulkcompensation at the receiver-side (typically performed in the digitaldomain). Dispersion pre-compensation at the transmit-side can be easilyincorporated via

⁰. The dispersion coefficient β₂−ps²/km and the nonlinearity coefficientis γ=1.1 W⁻¹ km⁻¹, both constant over z and ω. Additive noise due toamplified spontaneous emission (ASE) and laser PN are neglected since weonly focus on deterministic signal-signal NLI.

The numerical reference simulation is a full-vectorial field simulationimplemented via the symmetric split-step Fourier method [46] withadaptive step size and a maximum nonlinear phase-rotation per step ofϕ_(NL) ^(max)=3.5×10⁻⁴ rad. The simulation bandwidth is B_(sim)=8R_(s)for single-channel and 16R_(s) for dual-channel transmission. All filteroperations (i.e., pulse-shaping, linear step in the SSFM, linear channelmatched filter) are performed at the full simulation bandwidth via fastconvolution and regarding periodic boundary conditions.

The known fiber nonlinearity compensation schemes operating in thefrequency-domain are typically some sort of Volterra-based compensators(cf. [37,38,39]). All results following the Volterra approach operate ata fractional sampling rate (usually at two samples-per-symbol) and aretypically performed on the receive side (before linear equalization)jointly with (or instead of) chromatic dispersion compensation. Thoseapproaches hence do not incorporate the channel matched filter and donot establish and end-to-end relation between transmit and receivesymbol sequences. Those approaches also suffer from a higherimplementation complexity due to the higher sampling, i.e., processingrate and must run on the receive samples at a potentially high fixedpoint resolution. Run-time adaptation of the equalizer coefficients isalso hard to implement since the used control loop for the adaption ofthe coefficients has a long feedback cycle.

Derived from the frequency-domain description, a novel class ofalgorithms is provided which effectively compute the end-to-end relationbetween transmit and receive sequences over discrete frequencies fromthe (periodic) Nyquist interval. Remarkably, the frequency-matching in(31) which is imposed along with the general four wave mixing (FWM)process in the optical domain is still maintained in the periodicfrequency-domain.

For application in fiber nonlinearity compensation this scheme can bewell applied at the transmit-side during pulse-shaping (usually on thetransmit-side, pulse-shaping can be well combined with linearpre-compensation of transmitter components—typically done in thefrequency-domain anyway) or on the receive side after matched filtering.Moreover, while the time-domain implementation (cf. pulse collisionpicture) uses a triple summation per time-instance, the frequency-domainimplementation involves only a double summation per frequency index.Similar as for linear systems, this characteristic allows for veryefficient implementations using the fast Fourier transform when thetime-domain kernel comprises many coefficients. Since the proposedalgorithm only uses frequencies from within the Nyquist interval, it canbe implemented at the same rate as the symbol rate. In [35] it wasshown, that symbol pre-decisions (cf. decision-directed adaptation) canbe used to calculate the perturbative terms using the time-domainimplementation of the model. Symbol pre-decisions are also desirablesince they use only a low fixed-point resolution. Similarly, symbolpre-decisions can be used for the frequency-domain implementation (cf.symbol pre-decisions instead of the known symbols in Algorithm 1 and 2).

In the following, a discussion of the results is provided.

FIG. 7a and FIG. 7b illustrate contour plots of the normalizedmean-square error σ_(e) ²=E{∥y^(SSFM)-y^(PERT)∥²} in dB between theperturbation-based (PERT) end-to-end model and the split-step Fouriermethod (SSFM).

In particular, FIG. 7a illustrates a contour plot for a single-channel,single-span, lossless fiber scenario in the regular (REG) time-domain(TD) model (REG-PERT-TD) which is carried out as in (38).

FIG. 7b illustrates a contour plot for a single-channel, single-span,lossless fiber scenario in the regular-logarithmic (REGLOG) model(REGLOG-PERT-TD) which is carried out as in (55).

The results are shown w.r.t. the symbol rate R_(s) and the opticallaunch power of the probe P_(ρ)in dBm. Parameters as in Table I withroll-off factor ρ=0.2. N_(sp)=1, 10 log₁₀ e^(α)=0 dB/km and L_(an)=21.71km. a

In FIG. 7a , we start our evaluation with the most simple scenario,i.e., single-channel, single-span, and lossless fiber. The MSE is shownin logarithmic scale e in dB over the symbol rate R_(s) and the launchpower of the probe 10 log₁₀(P_(ρ)/mW) in dBm. The results are obtainedfrom the regular (REG) perturbation-based (PERT) end-to-end channelmodel in discrete time-domain (TD), corresponding to (38).

For the given effective length L_(eff) and dispersion parameter β₂, therange of the symbol rate between 1 GBd and 100 GBd corresponds to a mapstrength S_(T,ρ) between 0.003 and 28.7. This amounts to virtually nomemory of the intra-channel nonlinear interaction for small symbol rates(hence only very few coefficients h_(ρ)[k] exceeding the minimum energylevel of 10 log₁₀Γ^(SCI)=−60 dB) to a very broad intra-channel nonlinearmemory for high symbol rates (with coefficients h_(ρ)[k] covering alarge number of symbols). Likewise, the launch power of the probe P_(ρ)spans a nonlinear phase shift ϕNL.ρ from 0.02 to 0.34 rad. We canobserve a gradual increase in σ_(e) ² of about 5 dB per 1.5 dBm launchpower in the nonlinear transmission regime. We deliberately consider aMSE 10 log₁₀σ_(e) ²>−30 dB as a poor match between theperturbation-based model and the full-field simulation, i.e., here for Plarger than

$9{{dBm}\left( {\overset{\Delta}{=}{{0.168\mspace{14mu}{rad}} \approx {10{^\circ}}}} \right)}$

independent of R_(s).

TABLE I SIMULATION PARAMETERS a, b ∈ A PDM 64-QAM M$4096\mspace{14mu}\left( {\overset{\Delta}{=}{64\text{-}{QAM}\mspace{14mu}{per}\mspace{14mu}{polarization}}} \right)$h_(T)(t) h_(RRC)(t) with roll-off factor ρ γ 1.1 W⁻¹km⁻¹ β₂ −21 ps²/kmB₀ 0 ps² B(z) β₂z 10 log₁₀ e^(α) 0 dB/km 0.2 dB/km L_(sp) 21.71 km 100km

(z) 0 −αz + αLsp Σ_(i=1) ^(N) ^(sp) δ(z − iL_(sp)) N_(SYM) 2¹⁶ N_(DFT)max(

⁺¹, 64) 10 log₁₀ Γ −60 dB

In FIG. 7b the same system scenario is considered but instead of theregular model, now, the regular-logarithmic (REGLOG) model is employedaccording to (55). The gradual increase in σ_(e) ² with increasing P_(ρ)is now considerably relaxed to about 5 dB per 2.5 dBm launch power. Theregion of poor model match with 10 log₁₀e²>−30 dB is now only approachedfor launch powers larger than 12 dBm. We can also observe that σ_(e) ²improves with increasing symbol rate R_(s), in particular for ratesR_(s)>40 GBd. This is explained by the fact that the kernel energy E_(h)^(SCI) in (73) depends on the symbol rate R_(s), s.t. σ_(e) ² is reducedfor higher symbol rates.

FIG. 8a illustrates an energy of the kernel coefficients in atime-domain E_(h) over the symbol rate R_(s) (PERT-TD, single-channel,single-span).

FIG. 8b illustrates an energy of the kernel coefficients in afrequency-domain E_(H) over the symbol rate R_(s) (PERT-FD,single-channel, single-span).

The results are obtained from the regular-logarithmic (REGLOG) model fora single-channel (p=0.2) over a standard single-mode fiber (10log₁₀e^(α)=0.2 dB/km and L_(sp)=100 km) or a lossless fiber (10log₁₀e^(α)=0 dB/km and L_(sp)=21.71 km). The subscript Δ denotes thesubset of all coefficients associated with additive and the subscript ϕdenotes the subset of all coefficients with multiplicativeperturbations.

In particular, FIG. 8a shows the energy of the (time-domain) kernelcoefficients E_(h) ^(SCI) over R_(s) for a single-span SSMF withL_(sp)=100 km and for a lossless fiber with L_(sp)=21.71km.

Generally, we see that E_(h) ^(SCI) is constant for small R_(s) and thencurves into a transition region towards smaller energies before itstarts to saturate for large R_(s). For transmission over SSMF thistransition region is shifted to smaller R_(s), e.g., E_(h) ^(SCI) ropsfrom 0.7 to 0.6 around 33 GHz for lossless transmission and at around 20GHz for transmission over SSMF. We also present the kernel energiesE_(h,Δ) ^(SCI) associated with additive perturbations, and E_(h,ϕ)^(SCI) associated with multiplicative perturbations.

Most of the energy is concentrated in E_(h,ϕ) ^(SCI), i.e.,corresponding to the degenerate symbol combinations with k₁=0 or k₃=0defined in (58)-(60). Interestingly, while the total energy E_(h) ^(SCI)decreases monotonically with R_(s), the additive contribution E_(h)^(SCI) increases in the transition region and then decreases again forlarge R_(s). This behaviour is also visible in the results presented inFIG. 7 (a) and (b).

FIG. 8b shows the energy of the kernel coefficients E_(H) ^(SCI) infrequency-domain for the same system scenario as in (a). The totalenergies are the same (cf. Parseval's theorem), however, the majority ofthe energy is now contained in the regular, i.e., additive, subset ofcoefficients. Only, the amount of 1/N_(DFT) independent of R_(s) iscontained in the degenerate, i.e., multiplicative, subset ofcoefficients.

FIG. 9a illustrates a contour plot in the regular model in the frequencydomain of the normalized mean-square error σ_(e) ² in dB for asingle-channel, single-span, lossless fiber (REG-PERT-FD) according toan embodiment.

FIG. 9b illustrates a contour plot in the (₇₂ regular-logarithmic modelin the frequency domain of the normalized mean-square error σ_(e) ² indB for a single-channel, single-span, lossless fiber (REGLOG-PERT-FD)according to an embodiment.

The results are shown w.r.t. the symbol rate R_(s) and the opticallaunch power of the probe P_(ρ) in dBm. Parameters as in Table I withroll-off factor ρ=0.2, N_(sp)=1, 10 log₁₀ e^(α)=0 dB/km and L_(sp)=21.71km. In (a) the regular (REG) frequency-domain (FD) model is carried outas in Algorithm 1 and in (b) the regular-logarithmic (REGLOG) model iscarried out as in Algorithm 2.

In FIG. 9a and FIG. 9b the respective results on σ_(e) ² using thediscrete frequency-domain (FD) model according to Algorithm 1 and 2 areshown. We can confirm our previous statement that the regularperturbation model in time and frequency are equivalent considering thatthe results shown in FIG. 7a and FIG. 9b are (virtually) the same. Wealso conclude that the REGLOG-FD performs very similar to thecorresponding TD model despite the fact that only average terms cantruly be considered as multiplicative distortions. This may motivate theapplication of the FD over the TD model for fiber nonlinearitymitigation when an implementation in frequency-domain is computationallymore efficient.

FIG. 10a and FIG. 10b illustrate contour plots of the normalizedmean-square error σ_(e) ² in dB, wherein the results are obtained fromthe regular-logarithmic (REGLOG) time-domain (TD) model over a standardsingle-mode fiber with end-of-span lumped amplification (10log₁₀e^(α)=0.2 dB/km and L_(sp)=100 km).

In FIG. 10a , the symbol rate R_(s) and the optical launch power P_(ρ)are varied for single-span (N_(sp)=1) transmission and fixed roll-offfactor (σ=0.2). (REGLOG-PERT-TD, single-channel, single-span, standardfiber).

In FIG. 10b , the roll-off factor P and number of spans N_(sp) arevaried with fixed symbol rate (R_(s)=64 GBd) and fixed launch power (10log₁₀ (P_(ρ)/mW)=3 dBm). (REGLOG-PERT-TD, single-channel, multi-span,standard fiber).

The black cross in FIG. 10a and FIG. 10b indicates the point with acommon set of parameters. We can see a dependency on the roll-off factorp which is due to a dependency of E_(h) ^(SCI) on ρ (not shown here).With increasing β the kernel energy E_(h) ^(SCI) decreases and hencedoes σ_(e) ² too.

FIG. 10a and FIG. 10b show σ_(e) ² for a single-channel over standardsinglemode fiber (L_(sp)=100 km and 10 log₁₀e^(α)=0.2 dB/km) and lumpedend-of-span amplification. In the full-field simulation, the lumpedamplifier is operated in constant-gain mode compensating for the exactspan-loss of 20 dB. The results over a single-span in FIG. 10a areslightly better compared to the lossless case in FIG. 7b and thedependency on the symbol rate is even more pronounced. In FIG. 10b ,σ_(e) ² is shown over the roll-off factor σ and the number of spansN_(sp) for a fixed symbol rate of R_(s)=64 GBd and fixed launch power of10log₁₀(P_(σ)/mW)=3 dBm. The black cross in FIG. 10 (a) and (b) marksthe point with common set of parameters.

For dual-channel transmission the transmit symbols of the interferer

b[k]

are drawn from the same symbol set

. For both wavelength channels, the symbol rate is fixed to R_(s)=64 GBdand the roll-off factor of the RRC shape is σ=0.2. The transmit power ofthe probe is set to 10 log₁₀(p_(σ)/mW)=0 dBm while the transmit power ofthe interferer P₁ is varied together with the relative frequency offsetΔω/(2π) ranging from 76.8 GHz (i.e., no guard interval with 1.2×64 GHz)to 200 GHz.

In FIG. 11a , an energy of the kernel coefficients (black lines, bulletmarkers, left y-axis) in time-domain E_(h) over N_(sp) spans of standardsingle-mode fiber (10 log₁₀e^(α)=0.2 dB/km and L_(sp)=100 km, σ=0.2). isillustrated (PERT-TD, single-channel, multi-span, standard fiber).Additionally, the kernel energies are shown scaled with N_(sp)²′∞ϕ_(NL,σ) ² (gray lines, cross markers, right y-axis) to indicate thegeneral growth of nonlinear distortions with increasing N_(sp) (similarto the GN-model).

In FIG. 11b , kernel energies E_(h) for cross-channel interference (XCI)imposed by a single wavelength channel spaced at Δω₁/(2π) GHz over asingle span of lossless fiber. Both probe and interferer have R_(s)=64GBd and σ=0.2 are illustrated (PERT-TD, dual-channel, single-span,lossless fiber).

The scaling laws of σ_(e) ² with N_(sp) are complemented in FIG. 11a bythe energy of the kernel coefficients E_(h) ^(SCI) for the same systemscenario as in FIG. 10b (with σ=0.2). It is interesting to see that (forthis particular system scenario) E_(h,Δ) ^(SCI) and E_(h,ϕ) ^(SCI)intersect at N_(sp)=2. We can conclude that after the second span moreenergy is comprised within the additive subset of coefficients than inthe multiplicative one. With increasing N_(sp) the relative contributionof E_(h,Δ) ^(SCI) to the total energy E_(h) ^(SCI) is increasinp. Note,while E_(h) ^(SCI) is actually monotonically decreasing with N_(sp), thecommon prefactor ϕ_(NL,ρ) has to be factored in as it effectively scalesthe nonlinear distortion. Since for heterogeneous spans we haveϕ_(NL,ρ)∞L_(eff)∞N_(sp), the same traces are shown scaled by N_(sp) ² toillustrate how the energy of the total distortion accumulates withincreasing transmission length. In this respect, similar results can beobtained from the presented channel model as from the GN-model (givenproper scaling with ϕ_(NL,ρ) ² instead of just N_(sp) ², and similarlytaking all other wavelength channels into account). Additionally,qualitative statements can be derived, e.g., whether the nonlineardistortion is predominantly additive or multiplicative or on which timescale nonlinear distortions are still correlated.

FIG. 12a and FIG. 12b illustrate contour plots of the normalizedmean-square error σ_(e) ² in dB.

In particular, FIG. 12a illustrates a contour plot in a time domain, fordual-channel, single-span, lossless fiber, (REGLOG-PERT-TD).

FIG. 12b illustrates a contour plot in a frequency domain, fordual-channel, single-span, lossless fiber, (REGLOG-PERT-FD).

In FIG. 12a and FIG. 12b , the results are obtained from twoco-propagating wavelength channels with PDM 64-QAM and a symbol rate of64 GBd and roll-off factor σ=0.2. The launch power of the probe is fixedat 10 log₁₀(P_(ρ)/mW)=0 dBm while the power of the interferer P₁ and therelative frequency offset Δω₁ are varied. In (a) the regular-logarithmic(REGLOG) time-domain (TD) model for both SCI and XCI is carried out asin (55) and in (b) the REGLOG frequency-domain (FD) model is carried outas in Algorithm 2 and (78) for both SCI and XCI.

FIG. 12a and FIG. 12b show the e for dual-channel transmission using theREGLOG time-domain in FIG. 12a and the frequency-domain model in FIG.12b . The transmit symbols of the interferer

b[k]

are drawn from the same symbol set

. e.g., 64-QAM per polarization. For both wavelength channels, thesymbol rate is fixed to R_(s)=64 GBd and the roll-off factor of the RRCshape is σ=0.2. The transmit power of the probe is set to10log₁₀(P_(σ)/mW)=0 dBm while the transmit power of the interferer P_(v)with channel number v=1 is varied together with the relative frequencyoffset Δω₁/(2π) ranging from 76.8 GHz (i.e., no guard interval with(1+σ)×64 GHz) to 200GHz. In case of the end-to-end channel model bothcontributions from intra- and inter-channel distortions are combinedinto a single perturbative term (cf. (55) and (78)). The baseline errorσ_(e) ² is therefore approximately 55 dB considering the respective casewith R_(s)=64 GBd and P_(σ)=0 dBm in FIG. 7b . It is seen that the time-and frequency-domain model perform very similar. The dependency on thechannel spacing Δω₁ is explained considering FIG. 11b . Here, the energyof the cross-channel coefficients h₁[k] is shown over Δω¹. Generally,with increasing Δω₁. E_(h) ^(XCI) decreases and additionally therelative contribution of the degeneracy at k₃=0, i.e., E_(h) ^(SCI) isgrowing. Ultimately, the main distortion caused by an interferer spacedfar away from the probe channel is a distortion in phase and state ofpolarization.

Summarizing the above, a comprehensive analysis of end-to-end channelmodels for fiber-optic transmission based on a perturbation approach isprovided. The existing view on nonlinear interference following thepulse collision picture is described in a unified framework with a novelfrequency-domain perspective that incorporates the time-discretizationvia an aliased frequency-domain kernel. The relation between the time-and frequency-domain representation is elucidated and we show that thekernel coefficients in both views are related by a 3D discrete-timeFourier transform. The energy of the kernel coefficients can be directlyrelated to the GN-model.

While the pulse collision picture is a theory developed particularly forinter -channel nonlinear interactions, a generalization to intra-channel nonlinear interactions is presented. An intra-channel phasedistortion term and an intra-channel XPoIM term are introduced and bothcorrespond to a subset of degenerate intra-channel pulse collisions. Inanalogy to the time-domain model, the frequency-domain model is modifiedto treat certain degenerate mixing products as multiplicativedistortions. As a result, we have established a complete formulation ofstrictly regular (i.e., additive) models, and regular-logarithmic (i.e.,mixed additive and multiplicative) models, both in time- and infrequency-domain, both for intra- and inter-channel nonlinearinterference.

Provided from the frequency-domain description, a novel class ofalgorithms is implemented which effectively computes the end-to-endrelation between transmit and receive sequences over discretefrequencies from the Nyquist interval. In fiber nonlinearitycompensation this scheme can be well applied at the transmit-side duringpulse-shaping or on the receive side after matched filtering. Moreover,while the time-domain implementation uses a triple summation pertime-instance, the frequency-domain implementation involves only adouble summation per frequency index. Similar as for linear systems,this characteristic allows for very efficient implementations using thefast Fourier transform when the time-domain kernel comprises manycoefficients.

The provided algorithms were compared to the (oversampled and inherentlysequential) split-step Fourier method using the mean-squared errorbetween both output sequences. We show that, in particular, theregular-logarithmic models have good agreement with the split-stepFourier method over a wide range of system parameters. The presentedresults are further supported by a qualitative analysis involving thekernel energies to quantify the relative contributions of eitheradditive or multiplicative distortions.

In the following, a proof of the relation in (32), (33) is provided.

The Fourier transform of Δs(t) in (33) similarly computed as in [30,Appx.].

We start our derivation by expressing the optical field envelope u(0, t)by its inverse Fourier transform of U(0, ω) to obtain

$\begin{matrix}\begin{matrix}{{\Delta\;{s(t)}} = {{- j}\;\gamma\frac{8}{9}L_{eff}{\int_{{\mathbb{R}}^{2}}{{h_{NL}\left( {\tau_{1},\tau_{2}} \right)} \times}}}} \\{{u\left( {0,{t + \tau_{1}}} \right)}{u^{H}\left( {0,{t + \tau_{1} + \tau_{2}}} \right)}{u\left( {t + \tau_{2}} \right)}d^{2}\tau} \\{= {{- j}\;\gamma\frac{8}{9}L_{eff}\frac{1}{\left( {2\pi} \right)^{3}}{\int{\int_{- \infty}^{+ \infty}{d\tau_{1}d\tau_{2}{h_{NL}\left( {\tau_{1},\tau_{2}} \right)} \times}}}}} \\{\int_{- \infty}^{\infty}{d\omega_{3}{U\left( {0,\omega_{3}} \right)}{\exp\left( {j\omega_{3}\tau_{1}} \right)} \times}} \\{\int_{- \infty}^{\infty}{d\omega_{2}{U^{H}\left( {0,\omega_{2}} \right)}{\exp\left( {{- j}{\omega_{2}\left( {\tau_{1} + \tau_{2}} \right)}} \right)} \times}} \\{\int_{- \infty}^{\infty}{d\omega_{1}{U\left( {0,\omega_{1}} \right)}{\exp\left( {j\omega_{1}\tau_{2}} \right)} \times}} \\{{\exp\left( {{j\left( {\omega_{3} - \omega_{2} + \omega_{1}} \right)}t} \right)}.}\end{matrix} & (92)\end{matrix}$

The Fourier transform of the former expression yields

$\begin{matrix}\begin{matrix}{{\Delta{S(\omega)}} = {{- j}\;\gamma\frac{8}{9}L_{eff}\frac{1}{\left( {2\pi} \right)^{3}}{\int{\int{\int_{- \infty}^{+ \infty}{{dt}\; d\;\tau_{1}d\tau_{2}{h_{NL}\left( {\tau_{1},\tau_{2}} \right)} \times}}}}}} \\{\int_{- \infty}^{\infty}{d\omega_{3}{U\left( {0,\omega_{3}} \right)}{\exp\left( {j\omega_{3}\tau_{1}} \right)} \times}} \\{\int_{- \infty}^{\infty}{d\omega_{2}{U^{H}\left( {0,\omega_{2}} \right)}{\exp\left( {{- j}\omega_{2}\left( {\tau_{1} + \tau_{2}} \right)} \right)} \times}} \\{\int_{- \infty}^{\infty}{d\omega_{1}{U\left( {0,\omega_{1}} \right)}{\exp\left( {j\omega_{1}\tau_{2}} \right)} \times}} \\{{\exp\left( {{j\left( {\omega_{3} - \omega_{2} + \omega_{1} - \omega} \right)}t} \right)}.}\end{matrix} & (93)\end{matrix}$

We now use the identity ∫_(−∞) ^(∞exp(j(ω) ₃−ω₂+ω₁)t)dt=2πδ(ω₃−ω₂+ω₁−ω)to obtain

$\begin{matrix}\begin{matrix}{{\Delta\;{S(\omega)}} = {{- j}\;\gamma\;\frac{8}{9}L_{eff}\frac{1}{\left( {2\pi} \right)^{2}}{\int{\int_{- \infty}^{+ \infty}{d\tau_{1}d\tau_{2}{h_{NL}\left( {\tau_{1} \cdot \tau_{2}} \right)} \times}}}}} \\{{U\left( {0,{\omega - \omega_{1} + \omega_{2}}} \right)}{\exp\left( {{j\left( {\omega - \omega_{1} + \omega_{2}} \right)}\tau_{1}} \right)} \times} \\{\int_{- \infty}^{\infty}{d\omega_{2}{U^{H}\left( {0,\omega_{2}} \right)}{\exp\left( {{- j}{\omega_{2}\left( {\tau_{1} + \tau_{2}} \right)}} \right)} \times}} \\{\int_{- \infty}^{\infty}{d\omega_{1}{U\left( {0,\omega_{1}} \right)}{{\exp\left( {j\omega_{1}\tau_{2}} \right)}.}}}\end{matrix} & (94)\end{matrix}$

After re-arranging the order of integration, we have

$\begin{matrix}\begin{matrix}{{\Delta\;{S(\omega)}} = {{- j}\;\gamma\frac{8}{9}L_{eff}\frac{1}{\left( {2\pi} \right)^{2}}{\int{\int_{- \infty}^{+ \infty}{d\omega_{1}d\omega_{2} \times}}}}} \\{{U\left( {0,{\omega - \omega_{1} + \omega_{2}}} \right)}{U^{H}\left( {0,\omega_{2}} \right)}{U\left( {0,\ \omega_{1}} \right)} \times} \\{\int{\int_{- \infty}^{\infty}{d\tau_{1}d\tau_{2}{h_{NL}\left( {\tau_{1},\tau_{2}} \right)}{\exp\left( {j\omega_{1}\tau_{2}} \right)} \times}}} \\{{\exp\left( {{- j}{\omega_{2}\left( {\tau_{1} + \tau_{2}} \right)}} \right)}{{\exp\left( {{j\left( {\omega - \omega_{1} + \omega_{2}} \right)}\tau_{1}} \right)}.}}\end{matrix} & (95)\end{matrix}$

And finally a change of variables with υ₁=ω₁−ω and υ₂=ω₂−ω₁ yields

$\begin{matrix}\begin{matrix}{{\Delta\;{S(\omega)}} = {{- j}\;\gamma\frac{8}{9}L_{eff}\frac{1}{\left( {2\pi} \right)^{2}}{\int{\int_{- \infty}^{+ \infty}{dv_{1}dv_{2} \times}}}}} \\{{U\left( {0,{\omega + v_{2}}} \right)}{U^{H}\left( {0,{\omega + v_{1} + v_{2}}} \right)}{U\left( {0,{\omega + v_{1}}} \right)} \times} \\{\underset{{H_{NL}{({v_{1},v_{2}})}} = {\mathcal{F}{\{{h_{NL}{({\tau_{1},\tau_{2}})}}\}}}}{\underset{︸}{{\int{\int_{- \infty}^{+ \infty}{d\;\tau_{1}d\;{\tau_{2}\left( {\tau_{1},\tau_{2}} \right)}{\exp\left( {{{- {jv}_{1}}\tau_{1}} - {jv_{2}\tau_{2}}} \right)}}}},}}}\end{matrix} & (96)\end{matrix}$

which is equivalent to the expression in (32).

Although some aspects have been described in the context of anapparatus, it is clear that these aspects also represent a descriptionof the corresponding method, where a block or device corresponds to amethod step or a feature of a method step. Analogously, aspectsdescribed in the context of a method step also represent a descriptionof a corresponding block or item or feature of a correspondingapparatus. Some or all of the method steps may be executed by (or using)a hardware apparatus, like for example, a microprocessor, a programmablecomputer or an electronic circuit. In some embodiments, one or more ofthe most important method steps may be executed by such an apparatus.

Depending on certain implementation requirements, embodiments of theinvention can be implemented in hardware or in software or at leastpartially in hardware or at least partially in software. Theimplementation can be performed using a digital storage medium, forexample a floppy disk, a DVD, a Blu-Ray, a CD, a ROM, a PROM, an EPROM,an EEPROM or a FLASH memory, having electronically readable controlsignals stored thereon, which cooperate (or are capable of cooperating)with a programmable computer system such that the respective method isperformed. Therefore, the digital storage medium may be computerreadable.

Some embodiments according to the invention comprise a data carrierhaving electronically readable control signals, which are capable ofcooperating with a programmable computer system, such that one of themethods described herein is performed.

Generally, embodiments of the present invention can be implemented as acomputer program product with a program code, the program code beingoperative for performing one of the methods when the computer programproduct runs on a computer. The program code may for example be storedon a machine readable carrier.

Other embodiments comprise the computer program for performing one ofthe methods described herein, stored on a machine readable carrier.

In other words, an embodiment of the inventive method is, therefore, acomputer program having a program code for performing one of the methodsdescribed herein, when the computer program runs on a computer.

A further embodiment of the inventive methods is, therefore, a datacarrier (or a digital storage medium, or a computer-readable medium)comprising, recorded thereon, the computer program for performing one ofthe methods described herein. The data carrier, the digital storagemedium or the recorded medium are typically tangible and/ornon-transitory.

A further embodiment of the inventive method is, therefore, a datastream or a sequence of signals representing the computer program forperforming one of the methods described herein. The data stream or thesequence of signals may for example be configured to be transferred viaa data communication connection, for example via the Internet.

A further embodiment comprises a processing means, for example acomputer, or a programmable logic device, configured to or adapted toperform one of the methods described herein.

A further embodiment comprises a computer having installed thereon thecomputer program for performing one of the methods described herein.

A further embodiment according to the invention comprises an apparatusor a system configured to transfer (for example, electronically oroptically) a computer program for performing one of the methodsdescribed herein to a receiver. The receiver may, for example, be acomputer, a mobile device, a memory device or the like. The apparatus orsystem may, for example, comprise a file server for transferring thecomputer program to the receiver.

In some embodiments, a programmable logic device (for example a fieldprogrammable gate array) may be used to perform some or all of thefunctionalities of the methods described herein. In some embodiments, afield programmable gate array may cooperate with a microprocessor inorder to perform one of the methods described herein. Generally, themethods may be performed by any hardware apparatus.

The apparatus described herein may be implemented using a hardwareapparatus, or using a computer, or using a combination of a hardwareapparatus and a computer.

The methods described herein may be performed using a hardwareapparatus, or using a computer, or using a combination of a hardwareapparatus and a computer.

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which will beapparent to others skilled in the art and which fall within the scope ofthis invention. It should also be noted that there are many alternativeways of implementing the methods and compositions of the presentinvention. It is therefore intended that the following appended claimsbe interpreted as including all such alterations, permutations, andequivalents as fall within the true spirit and scope of the presentinvention.

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1. An apparatus for determining an interference in a transmission medium during a transmission of a data input signal, comprising: a transform module configured to transform the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to acquire a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient of the plurality of spectral coefficients is assigned to one of the plurality of frequency channels, and an analysis module configured to determine the interference by determining one or more spectral interference coefficients, wherein each of the one or more spectral interference coefficients is assigned to one of the plurality of frequency channels, wherein the analysis module configured to determine each of the one or more spectral interference coefficients depending on the plurality of spectral coefficients and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the two or more argument values indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values.
 2. The apparatus according to claim 1, wherein the transmission medium is a fiber-optical channel.
 3. The apparatus according to claim 1, wherein the apparatus further comprises a signal modification module being configured to modify the frequency-domain data signal using the one or more spectral interference coefficients to acquire a modified data signal, wherein the apparatus further comprises an inverse transform module configured to transform the modified data signal from the frequency domain to the time domain to acquire a corrected time-domain data signal.
 4. The apparatus according to claim 3, wherein the signal modification module is configured to combine each one of the one or more spectral interference coefficients, or a value derived from said one of the one or more spectral interference coefficients, and one of the plurality of spectral coefficients to acquire the modified data signal.
 5. The apparatus according to claim 3, wherein the transform module is configured to transform the data input signal from the time domain to the frequency domain by transforming a plurality of overlapping blocks of the data input signal from the time domain to the frequency domain to acquire a plurality of blocks of the frequency-domain data signal, and wherein the inverse transform module configured to transform the modified data signal from the frequency domain to the time domain by transforming a plurality of blocks from the frequency domain to the time domain and by overlapping said plurality of blocks being represented in the time domain to acquire the corrected time-domain data signal.
 6. The apparatus according to claim 1, wherein the apparatus further comprises an inverse transform module configured to transform the one or more spectral interference coefficients from the frequency domain to the time domain, and wherein the apparatus further comprises a signal modification module being configured to modify the data input signal being represented in the time domain using the one or more spectral interference coefficients being represented in the time domain to acquire a corrected time-domain data signal.
 7. The apparatus according to claim 6, wherein the signal modification module is configured to combine each one of the one or more spectral interference coefficients being represented in the time domain, or a value derived from said one of the one or more spectral interference coefficients, and a time domain sample of a plurality of time domain samples of the data input signal being represented in the time domain to acquire the corrected time-domain data signal.
 8. The apparatus according to claim 6, wherein the transform module is configured to transform the data input signal from the time domain to the frequency domain by transforming a plurality of overlapping blocks of the data input signal from the time domain to the frequency domain to acquire a plurality of blocks of the frequency-domain data signal, and wherein the inverse transform module is configured to transform a plurality of interference coefficients blocks from the frequency domain to the time domain, said plurality of blocks comprising the one or more spectral interference coefficients, and wherein the signal modification module is configured to modify the overlapping blocks of the data input signal, being represented in the time domain, using the plurality of interference coefficients blocks to acquire a plurality of corrected blocks, wherein the signal modification module is configured to overlap the plurality of corrected blocks to acquire the corrected time-domain data signal.
 9. The apparatus according to claim 3, wherein the apparatus further comprises a transmitter module configured to transmit the corrected time-domain data signal over the transmission medium.
 10. The apparatus according to claim 3, wherein the apparatus further comprises a receiver module configured to receive the data input signal being transmitted over the transmission medium.
 11. The apparatus according to claim 1, wherein the analysis module is configured to determine an estimation of a perturbated signal depending on the data input signal using the one or more spectral interference coefficients.
 12. The apparatus according to claim 11, wherein the analysis module is configured to determine the estimation of the perturbated signal by adding each one of the one or more spectral interference coefficients with one of the plurality of spectral coefficients.
 13. The apparatus according to claim 1, wherein each of the two or more argument values is a channel index being an index which indicates one of the plurality of frequency channels, or wherein each of the two or more argument values is a frequency which indicates one of the plurality of frequency channels, wherein said one of the plurality of frequency channels comprises said frequency.
 14. The apparatus according to claim 1, wherein the analysis module is configured to determine each spectral interference coefficient of the one or more spectral interference coefficients by determining a plurality of addends, wherein the analysis module is configured to determine each of the plurality of addends as a product of three or more of the spectral coefficients and of the return value of the transfer function, the transfer function comprising three or more channel indices or three or more frequencies as the two or more argument values of the transfer function, which indicate three or more of the plurality of frequency channels to which said three or more of the spectral coefficients are assigned.
 15. The apparatus according to claim 14, wherein the analysis module is configured to determine the interference by applying a regular perturbation approach.
 16. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: ${\delta\;{A_{\lambda}^{SCI}\lbrack\mu\rbrack}} = {{- j}\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}} \times {\sum_{\mu_{1},\mu_{2}}{{A_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{A_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$ wherein ΔA_(λ) ^(SCI)[μ] is said spectral interference coefficient, wherein A_(λ)[μ₁] is a first one of the three or more spectral coefficients, wherein A_(λ)[μ₂] is a second one of the three or more spectral coefficients, wherein, A_(λ)[μ₃] is a third one of the three or more spectral coefficients, wherein ^(H) indicates Hermitian, wherein μ₁ is a first index which indicates a first one of the plurality of frequency channels, wherein μ₂ is a second index which indicates a second one of the plurality of frequency channels, wherein μ₃ is a third index which indicates a third one of the plurality of frequency channels, wherein H_(ρ)[μ₁, μ₂, μ₃] indicates the transfer function, wherein N_(DFT) ² indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ_(NL,ρ) is a number.
 17. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: ${\Delta{A_{\lambda}^{XCI}\lbrack\mu\rbrack}} = {{- j}\frac{\phi_{{NL},\nu}}{N_{DFT}^{2}} \times {\sum_{\mu_{1},\mu_{2}}{\left( {{{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}} + {{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}I}} \right) \times {A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\nu}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$ wherein ΔA_(λ) ^(XCI)[μ] is said spectral interference coefficient, wherein B_(λ)[μ₁] is a first one of the three or more spectral coefficients, wherein B_(λ)[μ₂] is a second one of the three or more spectral coefficients, wherein, A_(λ)[μ₃] is a third one of the three or more spectral coefficients, wherein I indicates an identity matrix, wherein ^(H) indicates Hermitian, wherein μ₁ is a first index which indicates a first one of the plurality of frequency channels, wherein μ₂ is a second index which indicates a second one of the plurality of frequency channels, wherein μ₃ is a third index which indicates a third one of the plurality of frequency channels, wherein H_(v), [μ₁, μ₂, μ₃ ] indicates the transfer function, wherein N_(DFT) ² indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ_(NL,v) is a number.
 18. The apparatus according to claim 14, wherein the analysis module is configured to determine the interference by applying a regular logarithmic perturbation approach.
 19. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: ${\Delta{A_{\lambda}^{SCI}\lbrack\mu\rbrack}} = {{- j}\frac{\phi_{{NL},\rho}}{N_{DFT}^{2}} \times {\sum_{u}{{A_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{A_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\rho}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$ wherein ΔA₂ ^(XCI)[μ] is said spectral interference coefficient, wherein A_(λ)[μ₁] is a first one of the three or more spectral coefficients, wherein A_(λ)[μ₂] is a second one of the three or more spectral coefficients, wherein, A_(λ)[μ₃] is a third one of the three or more spectral coefficients, wherein ^(H) indicates Hermitian, wherein μ₁ is a first index which indicates a first one of the plurality of frequency channels, wherein μ₂ is a second index which indicates a second one of the plurality of frequency channels, wherein μ₃ is a third index which indicates a third one of the plurality of frequency channels, wherein H_(ρ)[μ₁,μ₂, μ₃] indicates the transfer function, wherein N_(DFT) ² indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ_(NL,v) is a number, wherein

={[μ₁, μ₂]|μ₂≠μ₁Λμ₂≠μ₃}.
 20. The apparatus according to claim 14, wherein the analysis module is configured to determine each spectral interference coefficient depending on: ${\Delta{A_{\lambda}^{XCI}\lbrack\mu\rbrack}} = {{- j}\frac{\phi_{{NL},\nu}}{N_{DFT}^{2}} \times {\sum_{u}{\left( {{{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}} + {{B_{\lambda}^{H}\left\lbrack \mu_{2} \right\rbrack}{B_{\lambda}\left\lbrack \mu_{1} \right\rbrack}I}} \right) \times {A_{\lambda}\left\lbrack \mu_{3} \right\rbrack}{H_{\nu}\left\lbrack {\mu_{1},\mu_{2},\mu_{3}} \right\rbrack}}}}$ wherein ΔA_(λ) ^(XCI)[μ] is said spectral interference coefficient, wherein B_(λ)[μ₁] is a first one of the three or more spectral coefficients, wherein B_(λ)[μ₂] is a second one of the three or more spectral coefficients, wherein, A_(λ)[μ₃] is a third one of the three or more spectral coefficients, wherein I indicates an identity matrix, wherein ^(H) indicates Hermitian, wherein t_(h) is a first index which indicates a first one of the plurality of frequency channels, wherein μ₂ is a second index which indicates a second one of the plurality of frequency channels, wherein μ₃ is a third index which indicates a third one of the plurality of frequency channels, wherein H_(v), [μ₁, μ₂, μ₃] indicates the transfer function, wherein N_(DFT) ² indicates a square of a number of the plurality of frequency channels of the frequency domain, wherein ϕ_(NL,v) is a number, wherein

={[μ₁, μ₂]^(T)|μ₂≠μ₁}.
 21. The apparatus according to claim 1, wherein the transfer function is normalized and nonlinear.
 22. The apparatus according to claim 1, wherein the analysis module is configured to employ Volterra based compensation to determine the one or more spectral interference coefficients.
 23. The apparatus according to claim 1, wherein the analysis module is configured to determine the one or more spectral interference coefficients by determining one or more transmit and receive sequences over discrete frequencies from a periodic Nyquist interval.
 24. The apparatus according to claim 1, wherein the transfer function depends on ${{H_{\nu}\left( e^{j\omega T} \right)} = {\frac{1}{T^{3}}{\sum\limits_{m \in Z^{3}}{H_{\nu}\left( {\omega - \frac{2\pi m}{T}} \right)}}}}.$
 25. A method for determining an interference in a transmission medium during a transmission of a data input signal, comprising: transforming the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to acquire a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient of the plurality of spectral coefficients is assigned to one of the plurality of frequency channels, and determining the interference by determining one or more spectral interference coefficients, wherein each of the one or more spectral interference coefficients is assigned to one of the plurality of frequency channels, wherein determining each of the one or more spectral interference coefficients is conducted depending on the plurality of spectral coefficients and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the two or more argument values indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values.
 26. A non-transitory digital storage medium having stored thereon a computer program for performing a method for determining an interference in a transmission medium during a transmission of a data input signal, comprising: transforming the data input signal from a time domain to a frequency domain comprising a plurality of frequency channels to acquire a frequency-domain data signal comprising a plurality of spectral coefficients, wherein each spectral coefficient of the plurality of spectral coefficients is assigned to one of the plurality of frequency channels, and determining the interference by determining one or more spectral interference coefficients, wherein each of the one or more spectral interference coefficients is assigned to one of the plurality of frequency channels, wherein determining each of the one or more spectral interference coefficients is conducted depending on the plurality of spectral coefficients and depending on a transfer function, wherein the transfer function is configured to receive two or more argument values, wherein each of the two or more argument values indicates one of the plurality of frequency channels, and wherein the transfer function is configured to return a return value depending on the two or more argument values, when said computer program is run by a computer. 